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2 i ( ) PDF I +α II II III A: IV B: V C: III V I, II III IV V III IV [email protected]

3 ii I +α (order estimation)

.............................................4.................................................... 3.5..................................................... 4 3 7 3.................................................. 7 3.................................................. 7 3.3................................................ 9 3.

4 iii II (grad), (div), (rot)

............................................... 53 8..................................................... 54 8.3..................................................... 55 II 57 9 58 9................................................... 58 9................................................. 59 9.

5 iv III A ( )

5..................................................... 9 8 8..................................................... 8....................................... 8.3........................................... 8.4.

6 v 3 δ δ 5 3. δ δ IV B ( ) ( ) ( )

....................................... 64 5.................................................. 66 5.3............................................... 66 5.4.................................................... 68 5.

7 V C ( )

................................................. 7 3.3..................................................... 8 3.4................................................ 8 3.5.................................................... 3 3.

8 I +α

9 3. / ( ) ) 3.4 mm 4. 3 mm.34 cm 4.3 cm 3, : ( ).. ± ± , : ( ).. ±..7.7 ±. ( ) ).cm.. π cm ( )

10 4. (order estimation) order estimation: a b a ( ) b. ( b (4. ) ) ) m gt, t / (s) ) l l kg H O (6 3 ) (5.6 ) ) 5 34 m/s 34 (m/s) 5 (s) 7 (m) (km).3.3. m, kg, s, A MKSA (K) (mol) (cd) SI m : m (/ ) 7 kg : kg ( ) s : ( 33 Cs) A : m 7 N A m, kg, s, A (C): [C] [A] [s] I dq dt

11 5 (N): [N] [kg] [m]/[s ] F ma ( x ) 5 P d T c 9 G 3 m 6 M 6 µ 3 k 9 n h p da 5 f.3. ) : λ (m) (J) λ (m) c (m/s) h (J s) ch/λ E hν hc/λ ) : m (kg) v (m/s) [J][N] [m] ([kg] [m]/[s ]) [m] [kg] [m ]/[s ] m v [J] mv (/)mv (/) 3) : T (s) R (m) G (N m /kg m 3 /s kg) M (kg) GMT R 3 3 ( ) ( ) GMT 4π R 3

12 6.4. ( ) (a) (l) (b) (m) (c) (n) (d) (o) (e) (p) ( ) (f) (q) (g) (r) (h) (s) (i) (t) ( ) (j) mol (u) C (k) (v). ( ) (a) (b) mol (c) mol g (d) (e) 3. m ( ), kg ( ), s( ), A ( ), K( ) (a) (e) (i) (b) (f) (j) (c) (g) (k) ( E) (d) ( B) (h) (l) 4. (a) (d) /(4πϵ ) (g) (b) (e) (h) ( ) (c) (f) (i) 5. (m) (c) 6.. mg ( ) (5m 5m.m) 7. cal/cm ( 5 km)

13 7.5. (a).4 (l) 9. 3 kg (b) 5 4 (m) 5 (c) 6 9 km (n) 3. 7 (d) km (o) (e) 4 4 km (p) 5 Hz (f) Å m (q) 5 5 T (g) 3 kg (r).6 9 C (h) 4 kg (s) cal/g C (i) 7 kg (t). 3 hpa. 5 Pa. 5 N/m (j) 3 g (u) 3.4 m/s (k).7 7 kg (v) 9.8 m/s. (a) (b) (c) g (d) g (e) M kg M M (a) m (e) kg (i) s (b) A (f) J N m kg m /s (j) kg m (c) Hz /s (g) J/(kg K) (k) F qe [F/q] N/(As) (d) F q v B (h) V (l) C( ) [F/q v] W( ) IV A s N/(C (m/s)) [V] kg m /A s 3 kg /As Q CV [C] A s 4 /(kg m ) 4. (a) 6.7 [N m /kg ] (f) [J s] (b) 4. [J/cal] (g) 3. 8 [m/s] (c).4 3 [J/K] (h) [C/mol] (d) 9. 9 [N m /C ] (i) 8.3 [J/(mol K)] (e) 6. 3 [/mol]

(a) m (e) kg (i) s (b) A (f) J N m kg m /s (j) kg m (c) Hz /s (g) J/(kg K) (k) F qe [F/q] N/(As) (d) F q v B (h) V (l) C( ) [F/q v] W( ) IV A s N/(C (m/s)) [V] kg

14 [m] kg [c] m/s [ ] J kg m /s [mc ] kg m /s mc ( ) m E mc. mg. 6 kg c 3. 8 m/s (5m 5m m) mc (. 6 ) (3. 8 ) 9. [J] (5 ) (5 ) (. )..5 9 [g]. cal/(g K) 4. J/(g K) 9 4 [K] π ( ).83 7 [cm ] cal/min 4./6.4 [J/s] [J/s] ( ) E mc m 4 6 (3. 8 ) 4 9 [kg/s] ( 3 kg)

cal/(g K) 4. J/(g K) 9 4 [K] 4..5 9 4π (.5 8+3+ ).83 7 [cm ] cal/min 4.

15 9. a ( ), b, c θ a sin θ c a, cos θ b a, tan θ c b, cosecθ sin θ, sec θ cos θ, cot θ tan θ., a b + c, sin θ + cos θ, + tan θ cos θ. θ b c (rad) r θ h h θ h (rad) r ( ) 3 π 6, 45 π 4, 6 π 3, 9 π, 8 π O θ r (rad) π/6 π/4 π/3 π/ sin / / 3/ cos 3/ / / sin / 3 3. π/ xy ( ) x θ P P x y θ < π/ cos θ, sin θ

16 cos θ, sin θ P x y P y sinθ θ cos O θ x θ cos θ, sin θ tan θ tan θ sin θ cos θ sin( θ) sin θ, cos( θ) cos θ, tan( θ) tan θ, sin(θ + π ) cos θ, cos(θ + π ) sin θ, tan(θ + π ) tan θ, sin(θ + π) sin θ, cos(θ + π) cos θ, tan(θ + π) tan θ. tanθ cosθ.5 sin θ Π 3Π Π Π -.5 Π Π 3Π Π θ -

17 .3 sin(α+β) xy (x, y) α (x, y ) (x, y) (x, y ) ( x y ) ( cos α sin α sin α cos α ) ( x y ). β (x, y ) (x, y ) ( x y ) ( cos β sin β ) ( x ) sin β cos β y ( cos β ) ( sin β cos α sin α ) ( ) x sin β cos β sin α cos α y ( cos α cos β sin α sin β (sin α cos β + sin β cos α) sin α cos β + sin β cos α cos α cos β sin α sin β ) ( x y ). (x, y) α + β (x, y ) ( x y ) ( cos(α + β) sin(α + β) sin(α + β) cos(α + β) ( ) sin(α + β) sin α cos β + sin β cos α, ) ( x y cos(α + β) cos α cos β sin α sin β, tan(α + β) sin α cos β + sin β cos α tan α + tan β cos α cos β sin α sin β tan α tan β. sin θ sin(θ + θ) sin θ cos θ + sin θ cos θ sin θ cos θ, cos θ cos(θ + θ) cos θ cos θ sin θ sin θ cos θ sin θ cos θ sin θ, tan θ tan θ + tan θ tan θ tan θ tan θ tan θ. cos ), sin θ cos θ cos θ + cos θ tan θ cos θ + cos θ,, ( cos θ) sin θ sin θ ( + cos θ).

cos α cos α cos β sin α sin β ) ( x y ).

18 sin α cos β [sin(α + β) + sin(α β)], cos α cos β [cos(α + β) + cos(α β)];, sin α sin β [cos(α + β) cos(α β)]. α + β A, α β B sin A + sin B sin( A + B cos A + cos B cos( A + B P sin θ + Q cos θ ) cos( A B ), ) cos( A B ). [ ] P P + Q P + Q sin θ + Q P + Q sin θ P + Q [sin θ cos α + cos θ sin α] P + Q sin(θ + α) sin α Q P + Q, cos α P P + Q.

19 3.4., π 5, 7π 8 ( )., 75, 6 3. r θ r θ 4. sin π 6 sin π 4 sin π sin( θ) sin θ cos(θ + π ) sin θ tan(θ + π) tan θ 6. sin(θ π ) cos θ cos(θ π ) sin θ 7. sin 7π 8. cos 5π 6 9. tan π 8. sin, cos. sin(ωt) + sin(ωt). sin( π λ x) + sin[π (x + d)] λ

20 4.5. π (rad) 36 (rad) 36 π π 36 (rad) 5 π 7π 36 (rad) 8 π 8 π ( 57.3 ) π 5 7 7π 8 7. π (rad) 36 π 36 (rad) π π 36 (rad) (.75 (rad)) (rad) 5π (rad) 6 6 π 36 (rad) 6π 5 (rad) 3. r πr θ θ π πr θ π r θ 4. ABC A BC D ABD ADC 3 BAD CAD π 3 π 6 BDA CDA π π ADC AC BC DC AD AC DC sin π 6 DC AC, 3 AC sin π 3 AD AC 3 ABC AB AC ACB ABC π π 4 AC + AB BC AC BC π sin π 4 AC BC 4 A B B C π 4 π 3 A D π 6 π 3 C

21 5 5. (a) θ θ (x, y) (x, y) sin( θ) sin θ (b) θ θ + π (x, y) ( y, x) cos(θ + π ) sin θ (c) θ θ + π (x, y) ( x, y) tan(θ + π) tan θ y y y θ θ x θ + π θ x θ+π θ x (a) (b) (c) 6. θ θ π (x, y) (y, x) y sin(θ π ) cos θ cos(θ π x ) sin θ θ π θ sin 7π ( π sin 4 + π ) sin π 3 4 cos π 3 + cos π 4 sin π cos 5π 6 cos ( π 6 + π ) cos π θ π sin π 8 cos π 4 sin θ > cos θ >, cos π 8 + cos π 4 + tan π 8 sin π 8 cos π 8 + ( )( + ) + +. sin 3θ sin(θ + θ) sin θ cos θ + cos θ sin θ sin θ cos θ cos θ + ( cos θ ) sin θ (4 cos θ ) sin θ cos 3θ cos(θ + θ) cos θ cos θ sin θ sin θ ( sin θ) cos θ sin θ cos θ sin θ ( 4 sin θ) cos θ

22 6. sin(ωt) + sin(ωt) sin[ (ω + Ω) (ω Ω) t] cos[ t] ) Ω ω cos( t) sin[(ω )t] ω cos( t) ω. sin( π λ x) + sin[π λ (x + d)] sin [π λ x + π λ (x + d)] cos [π λ x π (x + d)] λ cos πd λ sin[π λ (x + d )] ) d n πd λ (n ) cos () () λ

23 7 3 3.,, 3,... n n a, a, a 3,..., a n,... {a n } a k {a n } k k a k k a 3 a n n, b n n, c n n {a n }, 4, 6,... {b n } {c n } : a, a + d, a + d, a + 3d,..., a + nd,... : a, ar, ar, ar 3,..., ar n,... : p, p, 3 p,..., n p, n a, a n+ a n + a ( )

24 3 8 : a a, a n+ a n + d : a a, a n+ ra n : a, a n+ [(a n ) /p + ] p a a, a n+ a n+ + a n a, a, a 3, a 4 3, a 5 5, a 7 8,... n a n [( ) n ( ) n ] 5 5 n,, 3 a n+ pa n + q (p ) (p ) a n+ c p(a n c) c pc q q c ( p) q q a n+ p{a n ( p) ( p) } q p {a n ( p) } p n q {a ( p) } a n p n q {a ( p) } + q ( p) pn a + ( pn ) q p a + (n )q (p ) a n+ pa n+ + qa n a n+ sa n+ r(a n+ sa n ) r + s p rs q r, s x px q (x r)(x s) a n+ sa n r(a n+ sa n ) r (a n sa n ) r n (a sa ) a n+ sa n r n (a sa )

25 {a n } p q (q p) Σ q a k a p + a p+ + + a q kp p ( p ) q ( ) n : a n n n S n k n k S n (n ) + n S n n + (n ) : a n a + (n )d S n (n + ) + (n + ) + + (n + ) + (n + ) n(n + ) S n n(n + ) n n a k [a + (k )d] na + k k n(n + ) d nd na + : a n r n a n n S n a k r k a a ( + r + + r n ) k k : a n n S n a ( + r + + r n ) rs n a ( r + + r n + r n ) n(n ) d ( r)s n a ( r n ) S n a ( r n ) r a ( r < n ) r (k + ) 3 k 3 3k + 3k + k n n [(k + ) 3 k 3 ] n [3k + 3k + ] k k { (n + ) 3 } { n 3 } n n n 3 k + 3 k + k k k (n + ) 3 n 3 k n(n + ) n k

26 3 n k k 3 : a n n a k k : a k n C k [ (n + ) 3 3 n(n + ) n k k(k + ) n k n(n + ) n ] ( k ) k + n(n + )(n + ) 6 ( ) + ( 3 ) + ( 3 4 ) + + ( n n + ) + ( + ) + ( ) + ( n ) n + (n ) n + n! k!(n k)! (n k,, n) n n a k nc k n k k n ( ) (x + y) n nc k x k y n k x y k π ( ) π π 4 π 6 π 8 k k k ( ) k ( k ) 7 + ( k ) 4 + ( (k + ) ) 7 + e.788 e k k! +! + 3! + 4! + 5! + γ.5775 γ lim n [ log e n + n k ] k

27 (a) 7 (b) cos x x (c). a n+ a n+ + a n, a a 3. {a n } {b n } a n [( ) n ( ) n ] 5 a n+ a n b n, a, b n+ 3b n, b a n, b n 4. (a) n (k ) (b) k n k (4k ) (c) n nc k ( ) k k 5. p (i) p (ii) n n + (a) 4 n n < n! (b) a, a n+ a n + n a n (n n + ) (c) n k k 3 n (n + ) 4

28 a a a 3 a 4 a 5 a n (a) n 5 (b) π/ 3π/ 5π/ 7π/ 9π/ (n )π/ (c) /4 /9 /6 /5 /n. a n+ a n+ + a n r, s x x x ± 5 a n sa n r n (a sa ) a n ra n s n (a ra ) r s ra n rsa n r n (a sa ) sa n sra n s n (a ra ) (r s)a n r n (a sa ) s n (a ra ) r + 5, s 5 r s 5 a sa s + 5 a ra r 5 r s a n [r n s n ] [( ) n ( ) n ] 5 3. b n b n+ 3b n, b b n 3(b n ) 3 ((b n ) 3 n (b ) b n 3 n ( ) + 3 n + a n a n + b n a n + (3 n + ) a n + (3 n 3 + ) + (3 n + ) a + (3 + ) + (3 n + ) + 3n 3 n + 3n + (n )

29 (a) (b) (c) n k n n n n(n + ) (k ) k n n k k k (4k ) n n k (k )(k + ) ( k k ) k + ( 3 ) + ( 3 ( 5 ) + n ) n + n + n nc k ( ) k [ + ( )] n k 5. (a) n < 4! 4 n 4 k k < k! n k + < k + k < k!(k + ) n k + 4 n n < n! (b) n a n (n n + ) a ( + ) n a n (n n + ) n k a n (n n + ) n k + a k+ a k + k (k k + ) + k (k + k + ) [(k + ) (k + ) + ] n k + n a n (n n + ) (c) n k 3 ( + ) n j k 4 n j + j+ k k 3 j k 3 + (j + ) 3 j (j + ) + (j + ) 3 k 4 4 (j + ) [j + 4(j + )] 4 (j + ) (j + ) n j + n

30 a ( ) n a n a a a (a n ) a n /a n a n a m (a n ) (a m ) a n + m a n+m a n a m a n m n m a n a n a n a n p a /p x p a a /p q a q/p r a r π s a s s {s n } a s n ( ) a s lim t s a t (t ). x a x x a a x a y a x+y a x a x (a x ) y a xy a e ( e n lim + ) n n e e e e / x a x x p a a /p e ikπ (k,,,... p ) k

31 e e x / (/) x sinh x ex e x, cosh x ex + e x, tanh x sinh x cosh x sinh cosh tanh 6 cosh x tanh x - sinh x -4 sinh x, cosh x, tanh x 4. y a x x y x a y x y y log a x a (a ) x r a x, s a y x log a r y log a s log a r + log a s x + y log a (a x+y ) log a (a x a y ) log a (r s). log a x + log a y log a (xy) ( ) log a log x a x log a (x y ) y log a x log a

32 4 6 x log a r y log a s t log r s r t s a y a y log a s log a r t t log a r tx log a s log a r y x t log r s e ln x log x a a x a x e x ln a a log a x log a x log e x/ log e a (/ ln a) ln x e ln x 4.3 f(x) x, y f(x + y) f(x)f(y) x y f() f( + ) {f()} f() n f(n) f(n )f() f(n )f()f() {f()} n f() f( n + n n ) f(n n )f( n ) {f( n )}n f() a (a > ) 4. g(x) g(x)g(y) ( g(x)g(y) (k + l n ) k n n k n n n! f(x) a x g(x) x k ) ( k! l x k y n k k!(n k)! k n y l ) l! x n n! k,l x k y l k! l! n! k!(n k)! xk y n k n n! (x + y)n g(x + y)

33 4 7 k + l n k, l g(x) g(x) e x e g() n ( + n) n n nc k ( k n )k + n n + n(n )! n n! n(n )(n ) + n 3! n + + n! 3 n! + +! ( n ) + 3! ( n )( n ) + + n! ( n )( n )... ( n n ) g(x ln a) e x ln a a x n n

34 , 8, , a n, m (a n ) m a nm ( 6. lim + n p,, 3, 4 n p n) 7. cosh x sinh x 8. sinh(x + y) sinh x cosh y + cosh x sinh y 9. 3 log.3, log log 4, log 5, log 6, log 8, log 9. log(x y ) y log x

35 , 8 56, ( 6 ) (65536) ( 3 ) ( ) ( 5) 7 4 ( 5) a n a n (a n ) m m (a n ) m (a a a) (a n ) (a a a) (a n ). (a a a) (a n ) a n m a nm 6. ( + ). ( + ) ( + 3) 3 ( + 4) 4 ( ) ( ) ( ) cosh x sinh x ( e x + e x ) ( e x e x ) 4 (ex + + e x ) 4 (ex + e x ) e x cosh x + sinh x, e x cosh x sinh x sinh(x + y) ex+y e (x+y) [ex e y e x e y ] (cosh x + sinh x)(cosh y + sinh y) (cosh x sinh x)(cosh y sinh y) [cosh x cosh y + sinh x cosh y + cosh x sinh y + sinh x sinh y cosh x cosh y + sinh x cosh y + cosh x sinh y sinh x sinh y] sinh x cosh y + cosh x sinh y

36 log 4 log log.3.6 log 5 log log log.3.7 log 6 log ( 3) log + log 3.78 log 8 log 3 3 log log 9 log 3 log log x s x e s x y (e s ) y e sy log(x y ) log(e sy ) sy ys y log x

37 ( ) ( ) a a R (a, b) (c, d) R R (c a, d b) y (c,d) ( ) R (c a) + (d b) (a,b) R O x 3 A (p, q, r) A n p + q + r n V (x, x,..., x n ) V x + x + + x n n x k k

38 a k k a k k a (a x, a y, a z ) k a (ka x, ka y, ka z ) k k k k a (a x, a y, a z ) b (b x, b y, b z ) ka a (k<) ka (k>) a + b (a x + b x, a y + b y, a z + b z ) a + b b a a b a+( ) b 5.4 a (a x, a y, a z ) b (b x, b y, b z ) a b a b a x b x + a y b y + a z b z (4. ) a a x + a y + a z a a a, b θ a b a b cos θ xy ( ) a (a x, a y, ) b (b x, b y, )

39 5 33 a x ϕ a b x ϕ b a x a cos ϕ a b x b cos ϕ b a y a sin ϕ a b y b sin ϕ b a b a x b x + a y b y a b (cos ϕ a cos ϕ b sin ϕ a sin ϕ b ) a b cos(ϕ a ϕ b ) a b cos(ϕ b ϕ a ) (5.) cos( x) cos x ϕ b ϕ a O b φb φ a a

40 ( ). q (a, b, c) d (k, l, m) 3. q (a, b, c) d (k, l, m) 4. a b a b cos θ a, b, c a b θ c a + b ab cos θ b c θ a 5. a (,, ), b (,, ) a b

41 O A, B P, Q, R OA a, OB b OQ k k k OQ k( OB + BA) k[ b + ( a b)] k ( a + b) AR l l OA + l AR a + l[ b a] ( l) a + l b OQ AR G k ( a + b) ( l) a + l b a, b k l /3 OG ( a + b)/3 BP m m OB + m BP b + m( a b) m a + ( m) b m /3 ( a + b)/3 OG A a P G Q O R b B. r r q + t d (t ) d r q O

42 r ( r q) d d r q q O r 4. a, b, c b a c b a a + b a b c a + b a b cos θ a b a b cos θ θ b c b - a a 5. a b θ a b a b cos θ a b (,, ) (,, ) ( ) + + a + + b ( ) + ( ) + cos θ / / θ π/3

43 x y e x (, ), e y (, ) θ e x, e y e x (cos θ, sin θ) e y ( sin θ, cos θ) a (a x, a y ) e x, e y a a x e x + a y e y a θ e y sinθ e y θ O cos θ sinθ θ e x e x cos θ a a x e x + a y e y a x (cos θ, sin θ) + a y ( sin θ, cos θ) (cos θ a x sin θ a y, sin θ a x + cos θ a y ) (a x, a y) ( a x a y ) ( cos θ ax sin θ a y sin θ a x + cos θ a y ) ( cos θ sin θ sin θ cos θ ) ( ax a y ) n m n m A A A 3 A A A 3 A 3 A 3 A 33 v v v 3 A v + A v + A 3 v 3 A v + A v + A 3 v 3 A k v + A k v + A 33 v 3 A v 3 (A v) j A ji v i i

44 A ka k (ka) ij ka ij A A A 3 A A A 3 A 3 A 3 A 33 n m A ± B (A ± B) ij A ij ± B ij A A A 3 A A A 3 A 3 A 3 A 33 ± ka ka ka 3 ka ka ka 3 ka 3 ka 3 ka 33 B B B 3 B B B 3 B 3 B 3 B 33 A ± B A ± B A 3 ± B 3 A ± B A ± B A 3 ± B 3 A 3 ± B 3 A 3 ± B 3 A 33 ± B 33 AB (AB) ij 3 A ip B pj p ( ) ( ) ( a b p q ap + br aq + bs c d r s cp + dr cq + ds ) ( ) ( ) ( ) p q a b pa + qc pb + qd r s c d ra + sc rb + sd AB BA 6.3 A ij i j I I, I ij δ ij δ ij δ ij { (i j ) ( )

45 6 39 n n n n A A AI IA A, (AI) ij A ik δ kj (IA) ij δ ik A kj A ij. k k A B I B A A AA A A I ( a b A c d ) ad bc A ( d b (ad bc) c a ) ad bc x, y ax + by p, cx + dy q ( ) a b A, x c d ( x y ), v ( p q ) A x v A A A A x A v x A v x 6.4 A A (deta ) ( ) a b A A ad bc c d A 3 3 A A A A 3 A A A 3 A 3 A 3 A 33 A A A A 33 + A A 3 A 3 + A 3 A A 3 A 3 A A 3 A A A 33 A A 3 A 3

46 (a) ( ) ( ) (b) (c). σ ( ), σ ( i i ), σ 3 ( ) [A, B] AB BA {A, B} AB + BA a) [σ, σ ] {σ, σ } b) [σ, σ 3 ] {σ, σ 3 } c) [σ 3, σ ] {σ 3, σ } ( ) a b 3. (a) A A (a + d)a + (ad bc)i O (O c d ) (b) a b c d A n (n ) ( ) cos θ sin θ 4. R(θ) R(θ) sin θ cos θ 5. 3x + 5y 7 x 9y 6. (a) cos θ sin θ r sin θ r cos θ (b) (c)

47 (a) ( ) (b) (c) ) v i k v k A ki. 3 [σ a, σ b ] iϵ abc σ c, {σ a, σ b } δ ab (a, b, c 3) c ϵ abc (abc) (3) ϵ abc (abc) (3) 3. (a) A ( ) ( ) ( a b a b a + bc ab + bd c d c d ac + cd bc + d (a + d)a ( a ) + ad ab + bd ac + cd ad + d (ad bc)i ( ad bc ) ad bc ) A (a + d)a + (ad bc)i O (b) a b c d A A + I O A A 4. R (θ) A n A A n A n A n n A n ( ( cos θ ( sin θ) cos θ ( sin θ) sin θ cos θ ) ) ( ) cos θ sin θ sin θ cos θ

48 ( ) ( x y ) ( 7 ) 6. (a) (b) (c) ( ( x y ) cos θ sin θ ) ( ( ( 9) 5 3 ) ( 7 ) ) ( ( ) ( r sin θ r cos θ cos θ(r cos θ) ( r sin θ) sin θ r cos θ + r sin θ r ) )

49 x f(x) f(x) (x, f(x)) (x + h, f(x + h)) f(x + h) f(x) h h f(x) x f(x) f(x+h) f(x) O x x+h df(x) dx lim f(x + h) f(x) h h df dx f (x) ( ) f f(x), g(x) d [f(x + h) ± g(x + h)] [f(x) ± g(x)] [f ± g] lim dx h h lim h [f(x + h) f(x)] ± [g(x + h) g(x)] h df dx ± dg dx d f(x + h)g(x + h) f(x)g(x) [f(x)g(x)] lim dx h h f(x + h)g(x + h) f(x)g(x + h) + f(x)g(x + h) f(x)g(x) lim h [ h ] f(x + h) f(x) f(g + h) g(x) lim g(x + h) + f(x) h h h df dx g + f dg dx d dx ( ) f g f(x+h) lim f(x) g(x+h) g(x) h h lim h h [ ] f(x + h)g(x) g(x + h)f(x) g(x + h)g(x)

50 7 44 lim [f(x + h)g(x) f(x)g(x) h g(x + h)g(x)h +f(x)g(x) g(x + h)f(x)] lim h g(x + h)g(x) g [ df dx g f dg dx [ f(x + h) f(x) g(x) h ] g(x + h) g(x) f(x) h g(x) c (c ) d dc df [cf(x)] f(x) + c dx dx dx c df dx f(x) f(x(t)) df dt x(t + h) x(t) a f(x(t + h)) f(x(t)) lim h h f(x(t + h)) f(x(t)) x(t + h) x(t) lim h x(t + h) x(t) h f(x + a) f(x) x(t + h) x(t) lim lim a a h h df dx dx dt ] 7. x n (n ) d (x + h) n x n dx xn lim h h lim h h [ lim h [nx n + (x n + nx n h + [ n ] lim h nc k x k h n k x n h k ] n(n ) x n h h n ) x n n(n ) x n h h n ] nx n x n (n ) x n x n x d dx d dx [(xn )(x n )] { } d dx (xn ) (x n ) + (x n ) d dx (x n ) nx n (x n ) + (x n ) d dx (x n ) d dx (x n ) nx n

51 7 45 sin h lim h h lim cos h h h OAB OA OB AOB h B OA OA C OC OD OB D OC OD cos h BC sin h OAB h/ OAB (/) sin h OCD (h/) cos h O B D C A h cos h sin h h h/ h + (h ) sin h lim h + cos h lim h + h sin h lim h + h h h a sin h sin( a) sin a sin h h (a > ) sin h lim h h sin a a sin a a cos h h ( + cos h) h lim h cos h cos h ( + cos h) lim h h h ( cos h) sin h lim lim h h h h lim h ( cos h) h lim h sin h h sin h d sin(x + h) sin(x) sin x cos h + cos x sin h sin x sin x lim lim dx h h h [ h lim sin x cos h + cos x sin h ] cos x h h h d cos(x + h) cos(x) cos x cos h sin x sin h cos x cos x lim lim dx h h h [ h lim cos x cos h sin x sin h ] sin x h h h d dx tan x d ( ) sin x cos x cos x sin x( sin x) dx cos x cos x cos x

52 7 46 f(x) ln x log e x e lim e lim x n ( + n) n n ( + x) x x /h [ ln lim h ( + h)(/h) ln( + h) a h a a lim a e a ] lim h ln( + h) h e a lim a a d e x+h e x dx ex lim h h d ln x lim dx h ln(x + h) ln x h e x e h lim e x h h lim h x ln( + h x ) h x x d dx ax d d dx log a x d dx dx ex ln a ln ae x ln a a x ln a ( ) ln x ln a x ln a x p (p ) y x p ln y p ln x x dy y dx p x dy dx y p x pxp 7.3 h f(x + h) f(x) + hf (x) f (x) f(x + h) > f(x) f(x + h) < f(x) x f(x) f (x) (y x 3 x ) x x

53 7 47 f(x) f (x) x x f (x) f (x) > f (x) < f(x) x a b f + + f f(a) f(b) y f(x) a b x

54 ω, h, v, g arctan x tan x (a) (d) (g) (j) d dx cos d x (b) dt esin t d (c) dt [e t sin t] ( ) d d (e) ln [ ] x d (f) [x ln x] dx tan x dx x + dx d sin(ωt) (h) d d sin(ωt) (i) dt [ arctan x dt [ dx d dt h + vt g t] d (k) dt h + vt g t] d (l) e dx x. P n (x) d n n n! dx n [(x ) n ] (n ) P (x), P (x), P (x) 3. U(r) a r b (r > a, b ) r r 4. f(x) n x n n! df dx 5. x < f(x) f(x) ( ) n x n df n dx + x n

55 (a) (b) (c) (d) (e) d dx cos x cos x(cos x) cos x sin x sin x d dt esin t e sin t cos t d dt [e t sin t] e t sin t + e t cos t e t ( sin t + cos t) ( ) d dx tan x [ ] d dx ln x x + (tan x) tan x cos x tan x sin x d [ ln x ln(x + ) ] dx x x x + x + x x(x + ) x x(x + ) (f) (g) (h) d dx [x ln x] ln x + x x ln x + d sin(ωt) ω cos(ωt) dt d dt sin(ωt) d dt [ω cos(ωt)] ω sin(ωt) (i) y arctan x tan y x x d dx tan y d dx x dy cos y dx dy dx cos y + tan y + x (j) (k) (l) d [h + vt g ] dt t v gt d [h dt + vt g ] t d [v gt] g dt d dx e x d dx [ xe x ] e x x( x)e x (x )e x

56 7 5. P (x)! (x ) P (x) d! dx (x ) x x P (x) d! dx [(x ) ] d 4 dx [(x )(x)] 8 4[(x ) + x(x)] (3x ) 3. U(r) d dr U(r) d [ a dr r b ] a ( b) (br a) r r 3 r r3 r a b r b a r b U + U U( a) b 4. f(x) n ( ) a U b x n n! ( a a b ) ( a b b ) a b 4a b b a b 4a df dx d [ ] + x + x dx! + xn n! x! + + xn (n )! + n x n (n )! n k k x k k! f(x) ( ) n 5. f(x) x n n n df dx d [ ] x x dx + x3 3 + ( )n x n + n x + x + ( ) k x k + ( ) n x n ( x) + x n

57 f(x) F (x) d F (x) f(x) dx F (x) f(x) ( ) f(x)dx F (x) F (x) x F (x + x) F (x) lim x x F (x + x) F (x) + f(x) x F (x + x) F (x + x) + f(x + x) x f(x) F (x) + [f(x) + f(x + x)] x. F (x + n x) F (x) + [f(x) + f(x + x) f(x + (n ) x)] x F (x + n x) F (x) [f(x) + f(x + x) f(x + (n ) x)] x f(x) x f(x + x) x... n x w ( ) x, n f x O x... f(t) x t x+n x x+w F (x + w) F (x) x+w x f(t)dt f f(t)dt x

58 8 5 F (x + w) F (x) ( F (x + w) F (x) ) f ( ) F d x f(t)dt f(x), dx a d b f(t)dt f(x) dx x d F (x) f(x) 7. dx 8.. f(x) x p (p ) x sin x cos x e x a x (a > ) f(x), g(x) b a F (x) ( ) (p + ) xp+ ln x cos x sin x e x ln a ax d df [fg] dx dx g + f dg dx [ d b df dx [fg]dx a dx g + f dg ] dx dx [ b df [fg] b a a dx g + f dg ] dx dx b a df b dx g dx [fg]b a a f dg dx dx [ ] b a [ ] b a W (x) [W (x)] b a W (b) W (a) ) π (sin x)xdx [( cos x)x] π π ( cos π)π ( cos ) π + + π cos xdx ( cos x) d dx xdx π ( cos x)dx π + [sin x] π π + sin π sin π

59 8 53 t a ln x dx t a ln x dx [x ln x]a t t t ln t a ln a a t d x ln x dx a dx x d t dx ln x dx t ln t a ln a a t a dx t ln t a ln a (t a) t a t ln t t ln t f(x) ln x F (x) x ln x x x ln x x x ln x x x dx 8..3 ) df dx f x t x(t) d df dx F (x(t)) dt dx dt f(x(t))dx dt t F f(x(t)) dx dt dt ) + x dx x tan θ x θ π/4 π/4 + x dx d (tan θ)dθ ( + tan θ) dθ π/4 cos π/4 θ cos θ dθ dθ π 4 π sin θ ) dθ ( r < ) + r cos θ + r t cos θ dt sin θ θ π t dθ π sin θ π + r cos θ + r dθ dt ( + r cos θ + r dθ )dθ + rt + r ( dt) + rt + r dt [ ln( + rt + r ) ] r r [ln( + r) ln( r) ] ( ) + r r ln r

60 (a) (c) (e) π x ln x dx sin xdx x n e x dx (n ) (b) (d) (f) π/ sin x cos x dx xe x dx dx x. n n. π/ ( ) n π/ sin n xdx sin n xdx n 3. x a + y (a, b ) b 4. r h

61 (a) (b) (c) (d) (e) I n x ln x dx π/ π [ x ln x ] sin x cos x dx sin x dx π x x dx x dx ( : lim x x ln x ) π/ [ ] x [ sin x dx ] π/ 4 cos x cos x dx [ x xe x dx [ e x ] x n e x dx (n ) I e x dx [ e x] 4 ] sin x π π I n [ x n ( e x ) ] nx n ( e x )dx n x n e x dx ni n I n ni n n(n )I n n!i n! (f) x sin θ dx cos θ x θ π/ dθ. I n π/ x sin n x dx I n π/ π/ sin θ cos θdθ π/ sin x sin n x dx [ ( cos x) sin n x ] π/ π/ + (n ) (n ) π/ I n + (n )I n (n )I n I n n n I n π/ cos x sin n x dx 4 π/ cos θ cos θdθ dθ π ( cos x)(n ) sin n x cos x dx ( sin x) sin n x dx (n )(I n I n )

62 y b 4. x a a b a x a dx x a sin t x a a t π/ π/ a b x π/ a a dx b sin t dx π/ π/ dt dt b cos t (a cos t) dt π/ π/ π/ ab cos + cos t t dt ab dt π/ π/ ab [t + ] π/ sin t πab ( : (. ) ) D D x ar cos t, y br sin t ( r, t π) ( x ) ( ) x J det r t a cos t ar sin t det abr b sin t br cos t D dxdy y r y t π π/ abrdtdr ab(π) rdr πab πab y d dy ( ) h y d r h πd dy π r h (h y) dy y h h π r h (h y) dy π r h [ (y h) 3 3 ] h πr h 3 h y d r

63 II

64 ( ) x t x x(t) O x(t) v(t) x t + t x(t + t) x(t + t) x(t) t t v(t) v(t) dx(t) dt x(t + t) x(t) lim t t v(t + t) v(t) t ( ) ( ) x(t+ t) x(t+ t) t x(t+ t) x(t) t t x(t + t) + x(t) x(t + t) ( t) t a(t) a(t) dv(t) dt F ma(t) m dv(t) dt m d x(t) dt m F x(t) x(t) (m) ( t : x(t ), v(t )) (F ) ( ) x(t)

65 t x(t) v(t) t + t t + t x(t + t) x(t) + v(t) t, v(t + t) v(t) + a(t) t v(t) + F m t x(t + t) x(t + t) + v(t + t) t x(t) + v(t) t + (v(t) + F ) m t t x(t) + v(t) t + F m ( t), v(t + t) v(t + t) + a(t + t) t v(t) + F m t + F m t t t (8.. ) ( : t t )

66 m t (a) : x vt (v ) (b) : z h gt (h t g ) (c) : z h + vt gt (v h, g ) (d) : (x, z) (x + v x t, z + v z t gt ) (t (x, y ), (v x, v z ) ) (e) : (x, y) R(cos(ωt), sin(ωt)) (R, ω ) (f) : x A sin(ωt + α) (A ω α ). m t z z(t) t z() z, dz/dt v g (a) (b) g ( ) dz(t) + mgz(t) (c) m dt 3. t d dt (x, y) (k ( cos(ωt)), k sin(ωt)) t t

67 (a) : dx dt v : d x dt (b) : dz dt gt : d x dt : md x dt g : md z dt mg (c) : dz dt v gt : d z dt g : z md dt mg (d) : d dt (x, z) (v x, v z gt) : : m d (x, z) m(, g) dt d (e) : (x, y) Rω( sin(ωt), cos(ωt)) dt d : dt (x, y) Rω (cos(ωt), sin(ωt)) : m d dt (x, y) ω (x, y) (f) : dx dt Aω cos(ωt + α) : d x dt : m d x dt ω x. (a) m d z dt mg d (x, z) (, g) dt Aω sin(ωt + α) (b) t : dz dt v gt : d z g dt ( ) ( ) m(v gt)( g) mgv + mgt (c) d m dt ( ) dz(t) + mgz(t) dt m dz(t) dt d z(t) dt + mg dz(t) dt mgv + mgt + mg(v gt) 3. (x(t), y(t)) t(k ( cos(ωs)), k sin(ωs))ds k(t ω sin(ωt), ω cos(ωt)) + (x, y ) (x(), y()) (, ) x, y k ω (x(t), y(t)) k (ωt sin(ωt), cos(ωt)) ω

68 6. f(x, x,..., x n ) f(x,..., x k + h,..., x n ) f(x,..., x k,..., x n ) f((x, x,..., x n ) lim x k h h d f(x,..., x n ) x k dx k f(x,..., x n ) df df f x dx + f x dx + f x n dx n n k f x k dx k n P n T V P nrt V (R ) dp P P P RT dn + dt + dv n T V V nr nrt dn + dt V V dv [ ] f(x, x,..., x n ) x, x,..., x n t, t,..., t n ; x i x i (t, t,..., t n ) t k (k,..., n) ( ) f t k n i f x i x i t k ) (x, y) (r, θ) ( r, θ π) x r cos θ, y r sin θ y r x + y, sin θ x + y, cos θ x x + y r x x x + y cos θ, sin θ y/ x + y x, y r y y x + y sin θ. cos θ θ x xy θ sin θ cos (x + y ) 3/ r cos θ θ y (x + y ) y / (x + y ) cos θ 3/ r,.

69 63 θ x sin θ r, θ y cos θ r f x f r r x + f θ θ x f f sin θ cos θ, r θ r f f r y r y + f θ θ y f f cos θ sin θ +. r θ r. 4 r r + dr θ θ + dθ dr R rdθdr + r dr, dθ θ r R θ π R dθ R R π r dθdr R ( π ) R r dθ dr π r dr πr f(x, x,..., x n ) dx dx... dx n x, x,..., x n k, k,..., k n x (k, k,..., k n ), x n (k, k,..., k n ),... x n (k, k,..., k n ) dx dx... dx n J dk dk... dk n x x x k k... k n x x x k J det k... k n.... x n k x n k... x n k n J J J 4

70 64 ) J det x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ ( θ π, ϕ π) x r y r z r x θ y θ z θ x ϕ y ϕ z ϕ det sin θ cos ϕ r cos θ cos ϕ r sin θ sin ϕ sin θ sin ϕ r cos θ sin ϕ r sin θ cos ϕ cos θ r sin θ r cos θ sin θ cos ϕ + r sin 3 θ sin ϕ ( r ) cos θ sin θ sin ϕ ( r ) sin 3 θ cos ϕ r cos θ sin θ + r sin 3 θ r sin θ ( R ) ) e x dx π 4π e x dx I (I > ) I ( ) I π. π x +y +z R dxdydz R π R ( π r sin θdθdr π r dr 4πR3 3 ) ( e x dx e r rdθdr π R π π R ) e y dy R r [ cos θ] π dr r sin θdϕdθdr e r rdr π e (x +y ) dxdy [ e r ]

71 65.3. (a) ln [ ] x y [ x +y (d) y xy x] (b) ln [ ] x x ] x +y (e) y [ y x (c) sin( x x + y ) ] (f) x [ y x. (x, y, x) x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ r, θ, r, θ, ϕ ϕ x, y, z 3. r x + y + z r [ ] x + y + z r 4. x a + y (a, b ) b (hint: x ar cos θ, y br sin θ ( r, θ π) ) 5. x a + y b + z (a, b, c ) c 6. < a < R, R < a R π π G, ρ, a G r + a ar cos θ ρr sin θdϕdθdr

72 66.4. (a) (b) (c) [ ] y ln x x + y [ ] x log x x + y y [ln x ln(x + y )] y x + y x [ln x ln(x + y )] x x x + y y x x(x + y ) x sin( x + y ) cos( x + y ) x x x + y cos( x + y ) x + y. (d) (e) (f) r θ ϕ x x r + y y r + z z r x x θ + y y θ + z z θ x x ϕ + y y ϕ + z z ϕ [ y xy x] x x x [ y y x] y x [ y x x] x [ ] y x y x 3 sin θ cos ϕ x + sin θ sin ϕ y + cos θ z, r cos θ cos ϕ x + r cos θ sin ϕ y r sin θ z, r sin θ sin ϕ x + r sin θ cos ϕ y 3. r x + y + z x r x (x + y + z ) / x [ (x + y + z ) 3/ (x) [ (x + y + z ) 3/ 3 (x + y + z ) 5/ (x)x ] ] r 3 + 3x r 5 y r r 3 + 3y r 5, z r r 3 + 3z r 5 [ ] x + y + z r 3 r + 3(x + y + z ) 3 r 5 3 r 3 + 3r r 5

73 67 4. D D x ar cos t, y br sin t ( r, t π) ( x ) ( ) x J det r t a cos t ar sin t det abr b sin t br cos t x r y r z r x θ y θ z θ D x ϕ y ϕ z ϕ dxdy y r y t π abrdtdr ab(π) rdr πab πab 5. D D x ar sin θ cos ϕ, y br sin θ sin ϕ, z cr cos θ ( r, θ π, ϕ π) a sin θ cos ϕ ar cos θ cos ϕ ar sin θ sin ϕ J b sin θ sin ϕ br cos θ sin ϕ br sin θ cos ϕ c cos θ cr sin θ abcr cos θ sin θ cos ϕ + abcr sin 3 θ sin ϕ + abcr sin θ cos θ sin ϕ + abcr sin 3 θ cos ϕ abcr (cos θ sin θ + sin 3 θ) abcr sin θ dxdydz 6. I I D R π π πgρ πgρ a πgρ a R R R π π 4πabc abcr sin θdϕdθdr πabc r dr 4π 3 abc G r + a ar cos θ ρr sin θdϕdθdr πgρ [ ] π r ar (r + a ar cos θ) / dr r { (r + a + ar) / (r + a ar) /} dr r( r + a r a )dr π R π r sin θdθdr r sin θ r + a ar cos θ dθdr r r R R < a r R < a r a a r I πgρ a R 4πGρR3 3a r{(r + a) (a r)}dr πgρ a ( G M ) (M 4πR3 a 3 ρ) R r dr < a < R I πgρ [ a R r{(r + a) (a r)}dr + r{(r + a) (r a)}dr a a πgρ [ a ] R r dr + ar dr πgρ [ a a a 3 a3 + a(r a )] 4πGρa 3 πgρ(r a ) πgρ( a 3 R ) ]

74 68. A B C B C C3 A θ r F ( r) F ( r) r F ( r) r F ( r) d r C r(t) (x(t), y(t), z(t)) t F ( r) d r F ( r(t)) d r C t dt dt ) ( µ) m (L, ) (, L) W y µmg L C F µmg d r d r C x -L O L F ( r) d r µmg d r d r µmg d r µmg d r d r dt dt C : r (L( t), ) ( t ) d r dt dt ( L, )dt Ldt W (C ) F d r µmgldt µmgl C C : r L(cos t, sin t) ( t π) d r π dt dt L( sin t, cos t)dt Ldt W (C ) F d r µmgldt µmglπ C C -L

75 69 s(t) (x(t), y(t)) (t t t ) C f(s)ds t t f(s(t)) ds dt dt, ) ds dt ( dx + dt ( ) dy dt (+ ) ) x + y r (r ) C : (x(t), y(t)) (r cos t, r sin t) ( t π) ) ( dx ds dx + dy + dt ( ) dy dt r dt sin t + r cos t dt rdt π ds rdt πr C. p, q R (θ, ϕ) ( θ π, ϕ < π) x R sin θ cos ϕ, x R sin θ sin ϕ, z R cos θ (x + y + z R ) z x dr q n dr p Σ y Σ r(p, q) f( r) Σ Σ r f( r) ds f( r(p, q)) p r q dpdq ) Q R Σ r r E Q r 4πϵ r n r r E n ds r ( ) Σ r θ (R cos θ cos ϕ, R cos θ sin ϕ, R sin θ), r ( R sin θ sin ϕ, R sin θ cos ϕ, ) ϕ r θ r ϕ (R sin θ cos ϕ, R sin θ sin ϕ, R sin θ cos θ)

76 7 Σ E n ds π π Q 4πϵ R R sin θ dθdϕ Q ϵ ( ) ( ) ( ) S A nds A dv V A d s ( A) n ds S

77 7.3. (a) (a, b) f(x, y) k x + y (k ) k (b) x (r, ) f(x, y) x + y (k ) µ (c) R f(x, y) π x + y (µ ). (P ) (V ) (T ) P V nrt (n R ) (a) A B C D P dv P P B C P A D O V V V (b) A B C P dv B C P P B P O A V V C V (c) (b) B C P V γ [ ] (γ ) A B C 3. a E E( r) k r S E nds ( n r/ r ) EdV S 4. a A A(x, y, z) (, ωx, ) (ω ) C A d s ( A) nds ( n (,, )) C V S

78 7.4. (a) C : (x(t), y(t)) (at, bt, ) (t ) ds a + b dt f(x, y)ds k (at) + (bt) a + b dt k(a + b ) tdt c k (a + b ). (b) C : (x(t), y(t)) (t,, ) (t r) ds dt r k f(x, y)ds c t dt k r (c) C : (x(t), y(t)) (R cos t, R sin t) (t π) ds Rdt π µ f(x, y)ds c π R cos t + R sin t Rdt µ (a) A B B C C D D A P RT V (V ) P ( ) V RT V (V ) P ( ) V P dv + P dv + + P dv V V P (V V ) + P (V V ) (P P )(V V ) (b) A B B C C A RT P V (V ) RT P ( ) V V RT P dv + V dv + V P dv RT ln( V ) + P (V V ) V V V RT ln( V V ) P (V V ) (c) (b) B C P V γ P V γ P V γ P P V γ P dv V V P V γ P V γ V dv P (V γ V ) V γ V γ γ P V P V γ V γ P (V V ) P (V V )

79 73 3. E n k r S r r k r ka E nds E k(x, y, z) 3k π π kaa sin θdθdϕ 4πka 3 V EdV a π π 3kr sin θdϕdθdr 3k 4πa3 3 4πka 3 4. C : (x(t), y(t)) (a cos t, a sin t) (t π) d s a( sin t, cos t, )dt C A s π ωa (, ωa cos t, ) a( sin t, cos t, )dt π + cos(t) dt ωπa π ωa cos dt A (,, ω) S ( A) nds a π ωrdθdr ωπa

80 74. f(x) (x a) n (a n ) ( ) c n f(x) c n (x a) n c + c (x a) + c (x a) + n () x a f(a) c () x f (x) + c + c (x a) + 3c 3 (x a) + x a f (a) c () x f (x) + c + 3 c 3 (x a) + 4 3(x a) x a f (a) c.. () x n dn dx n f(x) n (n ) c n + (n + ) n c n+ (x a) + x a f (n) (a) n!c n f(x) x a ( ) f(x) f(a) + f (a) (x a) + f (a) (x a) + f (a) 3! (x a) n n! f (n) (a) (x a) n. a f(x) f() + f () x + f () x + f () 3! x n n! f (n) () x n. ) e x + x + x + sin x x 3! x3 + cos x! x + n n n n! xn ( ) n (n + )! xn+ ( ) n (n)! xn

81 75 x ( ) x + x + x x n ( x < ) n ln( + x) x x + ( ) n 3 x3 + n n xn ( < x ) f(x, y) f(a, b) + f f (a, b)(x a) + (a, b)(y b) x y + f x (a, b)(x a) + f y (a, b)(y b) + f (a, b)(x a)(x b) +. xy. sin x, cos x x a sin, cos sin x cos x k k ( ) k (k + )! xk+ x x3 3! + x5 5! x7 7! +, ( ) k (k)! xk x! + x4 4! x6 6! +. sin x, cos x sin, cos ( ) e x k x k k! + x + x! + x3 3! +. x θ e iθ (iθ) k [ i m k k! m (m)! θm + ( ) m m (m)! θm + i m ] im+ (m + )! θm+ ( ) m (m + )! θm+ cos θ + i sin θ e iα e iβ (cos α + i sin α)(cos β + i sin β) (cos α cos β sin α sin β) + i(sin α cos β + cos α sin β) cos(α + β) + i sin(α + β) e i(α+β)

82 76 e A e B e A+B z, w e z e w k n e z+w z k k! n! j n k w j j! n k n k! (n k)! zk w n k (k + j n ) n! k!(n k)! zk w n k n! (z + w)n ( ) n

83 77.3. f(x) k k! f (k) () x k sin x, cos x. cos x x 4 x π 3 ( ) 3. x < x x x 4. a, r r a θ a + r ar cos θ r r 5. sin θ i (eiθ e iθ ) cos θ (eiθ + e iθ ) 6. N e ikx k

84 78.4. sin x, cos x n d dx sin x cos x, d sin x sin x,, dx d n { ( ) m dx n sin x sin x (n m ) ( ) m cos x (n m + ) d dx cos x sin x, d cos x cos x,, dx d n { ( ) m+ dx n cos x cos x (n m ) ( ) m+ sin x (n m + ). m x f (n) () ( ) m ( ) m sin x (m + )! xm+, cos x (m)! xm cos π 3 m ( π 3 m ) ( ) π 4 π + 4! π ( π 3 ) % 6! 3. π ( ) π 4 4! 3 x ( x) / f(x) f (x) (/) ( x) 3/ f (x) (3/4) ( x) 5/ z f() + f ()x +! f ()x + O(x 3 ) + x x + O(z 3 ) 4. a + r ar cos θ a [(a/r) cos θ (r/a) ] [(a/r) cos θ (r/a) ] x a + r ar cos θ [ + a x + 3 ] 8 x + O(x 3 ) [ + a [(a/r) cos θ (r/a) ] + 3 ] 8 [(a/r) cos θ (r/a) ] + O(x 3 ) [ + r r cos θ a a a + 3r cos ] θ + O(r 3 ) a a + r ( r 3 cos ) θ cos θ + + O(r 3 ) a a 3

85 79 5. e iθ cos θ + i sin θ e iθ cos( θ) + i sin( θ) cos θ i sin θ. sin θ i (eiθ e iθ ), cos θ (eiθ + e iθ ) 6. A N k e ikx A + e ix + e 4ix + e i(n )x (eix ) N einx e ix e ix einx (e inx e inx ) e ix (e ix e ix ) einx i sin(nx) e ix i sin x einx sin(nx) e ix sin x θ e iθ cosθ + i sin θ cos θ + sin θ A e inx sin(nx) e ix sin x sin (Nx) sin x ) N

86 f(x) (f (x), f (x), f (x)...,) F (x, f, f, f,...) f f n n : d f(x) kf(x) (k ) dx : m d dt x(t) mω x(t) η d x(t) (m, ω, η ) dt n n (f(x ) f, f (x ) f, f (x ) f,...) : ( ) 38 9U( ) α 34 9Th( ) ( ) 45 p t N(t) t t + t pn(t) t N(t) N(t + t) N(t) N(t) pn(t) t t t d N(t) pn(t) dt N(t) dn N dt p

87 3 8 t dn N dt dt ( p)dt N dn ( p)dt ln N(t) pt + C (C ) N(t) N(t )e p(t t ) C t t N(t ) N(t) N(t )e p(t t ) ( ) τ / e pτ / / p p ln /τ / (5 8 ) 3 6 : k x kx ( ) m m d x(t) kx(t) dt kx x x(t) A sin(ωt + B) (ω, A, B ) ω mω A sin(ωt + B) ka sin(ωt + B) k/m A, B 3. f(x, x,..., x n ) (x, y, z) t 3 3 r (x, y, z) x i x + y + z i ϕ( r) ρ( r) ( ρ( r) )

88 3 8 [ ] + µ ϕ( r) ρ( r) [ ] v t ψ( r, t) ρ( r, t) [ ] k t ψ( r, t) ρ( r, t) (v )

89 Ra Ra 6 88Ra /. d dt x(t) ω x(t) x(t) a) x() A, ẋ() b) x(), ẋ() v c) x() A, ẋ() v 3. v() g, k. dv(t) dt g kv(t) ( : e kt ) 4. f, g ξ(x, t) f(x vt) + g(x + vt) x ξ(x, t) v ξ(x, t) t

90 t ( ) t/(.6 3 ). t ln.6 3 ln t.6 3 ln ln d dt x(t) ω x(t) x(t) a sin(ωt + b) ẋ(t) aω cos(ωt + b) a b a) x() a sin b A, ẋ() aω cos b b π/, a A x(t) A sin(ωt + π/) A cos(ωt) b) x() a sin b, ẋ() aω cos b v b, a v/ω x(t) v ω sin(ωt) c) x() a sin b A, ẋ() aω cos b v ( ) A sin b + cos b + a ( v aω ) a A + (v/ω) x(t) A + (v/ω) sin(ωt + θ) θ sin θ A/ A + (v/ω), cos θ (v/ω)/ A + (v/ω) 3. e kt kt dv(t) e dt ge kt kv(t)e kt ge kt v(t) d dt {ekt } kt dv(t) e + d dt dt {ekt }v(t) ge kt d dt [ekt v(t)] ge kt t e kt v(t) g k ekt + C (C ) v() C g k ( ) v(t) g k + Ce kt v(t) g k [ e kt ]

91 x ξ(x, t) v ξ(x, t) ξ(x, t) f(x vt) t ξ(x, t) g(x + vt) x f(x vt) f (x vt) ( v) v f(x vt) f (x vt) f (x vt) t v x f(x vt) v f(x vt) t x g(x vt) g (x + vt) v g(x + vt) t f, f af + bf (a, b ) x [af + bf ] a x f + b x f a v t f + b v t f v t [af + bf ] af + bf ( : ) f(x vt), g(x + vt) f(x vt) + g(x + vt)

92 r (x, y, z) r ( ) : ( : ) : r ( : ) r x y z r r r 3 r x y z R x y z 3 r i R ij r j j, 3 3 RR T R T R r i (R T ) ik r k r kr ki k k (R T R : (R T ) ij R ji ) (R: ) : T ij...n (R ip )(R jq ) (R nu )T pq...u p q u ( : f xy ) 4. 3 a (a x, a y, a z ) b (b x, b y, b z ) ( ) a b a b (a y b z a z b y, a z b x a x b z, a x b y a y b x ) a b b a e x (,, ), e y (,, ), e z (,, )

93 4 87 e x e x e y e y e z e z e x e y e y e x e z y e e y e z e z e y e x y e z e z e x e x e z e y x e x a, b θ a b a b sin θ ( ) a (a x, a y, a z ) b (b x, b y, b z ) a b (a y b z a z b y ) + (a z b x a x b z ) + (a x b y a y b x ) (a y b z ) + (a z b y ) + (a z b x ) + (a x b z ) + (a x b y ) + (a y b x ) a y a z b z b y a z a x b z b x a x a y b x b y (a x + a y + a z)(b x + b y + b z) a xb x a yb y a zb z a y a z b z b y a z a x b z b x a x a y b x b y (a x + a y + a z)(b x + b y + b z) (a x b x + a y b y + a z b z ) a b ( a b) a b a b cos θ a b sin θ θ π sin θ a b a b sin θ 3 v x v y v z v 3 v (v, v, v 3 ) ϵ ijk ( a b) i 3 j,k ϵ ijk a j b k : (i, j, k) (,, 3), (, 3, ), (3,, ) ϵ ijk : (i, j, k) (, 3, ), (,, 3), (3,, ) : a c a (c ) ( a c a) i c c 3 j,k 3 k,j 3 j,k ϵ ijk a j ca k c ϵ ik j a k a j 3 j,k ϵ ijk a j a k (j k, k j ) ( ϵ ij k ) a j a k ( a c a) i a c a r p r p z

94 a) (,, ) (,, ) b) (,, ) (,, ) c) (,, ) (,, ) d) (v, v, ) (,, B) e) 3 (F, F, F ) (r,, r) f) (, p, ) (x, y, z). m d r dt F (a) r (b) p r p ( ) F r (c) F r ( F ) r p A, B, C A ( B C) B ( C A) C ( A B) 5. 3 A, B, C A ( B C) ( A C) B ( A B) C 6. q B F q v B B (,, B) ( q m) r() (a,, ), v() (, v, ) 7. r() (a,, ), v() (, v, v z )

95 (a) (,, ) (b) (,, ) (c) (, 5, 3 ) (d) (vb, vb, ) (e) 3 (F r,, F r) (f) (pz,, px). (a) (b) r m d r dt r F d d r d p ( r p) p + r dt dt dt d r dt md r dt + r r md dt r r md dt r F (c) F r r F (b) d dt ( r p) r p 3. a b 3 3 a i a i R ij a j, a i b i R ij b j j j a b a b a b a ib i R ij a j R ik b k R ij R ik a j b k i i j k i j k δ jk a j b k a j b j j k j 4. A ( B C) A i ( ϵ ijk B j C k ) ϵ ijk A i B j C k i j k i j k ϵ kij A k B i C j ϵ ijk B i C j A k k i j i j k B ( C A) ϵ jki A j B k C i ϵ ijk C i A j B k j k i i j k C ( A B) A, B, C

96 ( A ( B C) ) A (B C B C ) A 3 (B 3 C B C 3 ) (A C + A 3 B 3 )B (A B + A 3 B 3 )C (A C + A C + A 3 B 3 )B (A B + A B + A 3 B 3 )C ( A C)B ( A B)C ( A ( B C) ) ( A C)B ( A B)C, ( A ( B C) ) A ( B C) ( A C) B ( A B) C 6. m d r dt q v B qb(v y, v x, ) d r dt d v dt m dv x dt qbv y, d v x dt m dv y dt qbv x, ( qb m ) v x 3 ( A C)B ( A B)C 3 m dv z dt qb m ω v x(t) α sin(ωt+β) v x () β v x (t) α sin(ωt) x(t) α ω cos(ωt)+γ x() a γ α ω a y m dv x dt qbv y v y ω dv x dt αω cos(ωt) α cos(ωt) ω v y () v α v γ a + α ω a + v ω v y y(t) v sin(ωt) + δ y() δ ω z dv z dt v z(t) η v z () δ z() z(t) x(t) v ω [ cos(ωt)] + a, y(t) v sin(ωt), z(t) ω 7. z dv z dt v z() v z v z (t) v z z() z(t) v z t x(t) v ω [ cos(ωt)] + a, y(t) v ω sin(ωt), z(t) v zt

97 : t r(t) : t r ( E( r, t) B( r, t)) : : 5. ( x, y, z x, x y z x ỹ z R x y z x, y, z y, (R) xx (R) xy (R) xz (R)ỹx (R)ỹy (R)ỹz (R) zx (R) zy (R) zz ) z x x x x + ỹ x ỹ + z x z (R) xx x + (R) ỹx ỹ + (R) zx z x y y x + ỹ y ỹ + z y z (R) xy x + (R) ỹy ỹ + (R) zy z x z z x + ỹ z ỹ + z z z (R) xz x + (R) ỹz ỹ + (R) zz z R T R R R I x y z R ( ) x ỹ z ( x, y, x y z ) z

98 (grad), (div), (rot) (f( r)) ( A( r) (A x ( r), A y ( r), A z ( r)) (grad) : gradf( r) f( r) (div) : ( x f, y f, ) z f div A( r) A( r) x A x + y A y + z A z (rot) : rot A( r) A( r) ( A) i 3 j,k ( y A z z A y, ϵ ijk ( ) j ( A) k 3 j,k z A x x A z, ϵ ijk j A k x A y ) y A x grad M r m V ( r) G Mm r r r grad ( gradv ( r) G Mm ) ( ) Mm G r x + y + z ( ( ) ( ) Mm Mm G, G x x + y + z y x + y + z ( x GMm (x + y + z ), y 3/ ( ) GMm r r r GMm r (x + y + z ), ]; z 3/ (x + y + z ) 3/, ( )) Mm G z x + y + z ) r V ( r) V ( r) ( ) r

99 5 93 div a ρ E( r) 4πkρ r ( 6 3 grad ) div div E( r) E( r) x E x + y E y + z E z 4πkρ 3 ( x x + y y + z) 4πkρ z r Q E( r) kq (x, y, z) div r r dive( r) E( r) ( ) kq x r x + ( ) kq 3 y r y + ( ) kq 3 z r z 3 kq r 3 3kQ r 5 x + kq r 3kQ 3 r 5 y + kq r 3kQ 3 r 5 z (r ) ( ) div E( r) r div ( ) j( r) r rot A( r) (,, µ I 4π ln(x + y )) z z l x + y ( ).5.5 l A rot rota( r) (A x, A y, A z ) ( y A z z A y, ( µ I 4π y ln(x + y ), z A x x A z, µ I 4π z A y ) y A x ) x ln(x + y ), µ I πl ( y l, x l, )

100 5 94 z I ( ) A rot ( B A) A grad ( ) y x rot z

101 r (x, y, z) r r r r (a) r (d) r (g) r (b) (/r) (e) ( r/r 3 ) (h) ( r) (c) ( r) (f) ( )r (i) ( r). V ( r) k r (k ) V ( r) 3. B k (x + y ) ( y, x, ) ( x + y k ) B 4. m (,, ), A m r k B r 3 (r ) k r (x, y, z), r r B A 5. r (x, y, z) f f 6. r (x, y, z) A ( A) 7. r (x, y, z) A ( A) ( A) A

102 (a) r ( x + y + z ) ( x x + y + z, y x + y + z, ) z r x + y + z r (b) ( ) r ( x x + y + z, y x + y + z, ) z x + y + z ( ) x (x + y + z ), y 3/ (x + y + z ), z 3/ (x + y + z ) 3/ r r 3 (c) (d) (e) (f) ( r) 3 r x x + y y + z z 3 ( r/r 3 ) x x (x + y + z ) + 3/ y (x + y + z ) 3x 3/ (x + y + z ) 5/ + (x + y + z ) 3y 3/ (x + y + z ) 5/ ( )r y (x + y + z ) 3/ + z + (x + y + z ) 3z 3/ (x + y + z ) 5/ 3 (x + y + z ) 3(x + y + z ) 3/ (x + y + z ) 3 5/ r 3r 3 r 5 ( ) x + y + z x + y + z x y x + y + z + z x x + y + z + y x + y + z x (x + y + z ) 3/ + x + y + z y (x + y + z ) 3/ + x + y + z z (x + y + z ) 3/ 3 r (x + y + z ) r 3 r z (x + y + z ) 3/ z x + y + z

103 5 97 (g) (h) r ( r) ( z y y z, x z z x, y x ) x y ( x x + y + z, y x + y + z, ) z x + y + z ( z y x + y + z y z x + y + z, x z x + y + z z x x + y + z, y x x + y + z x y x + y + z ) yz ( (x + y + z ) + yz 3/ (x + y + z ), 3/ xz (x + y + z ) + xz 3/ (x + y + z ), 3/ xy (x + y + z ) + xy 3/ (x + y + z ) ) 3/ (i) ( r). ( ) V ( r) k r k (x + y + z ) k(x, y, z) k r 3. B k ( y x + y, ) ( x x + y, k xy (x + y ) xy (x + y ) + ) 4. A k m r r 3 5. k ( y, x, ) r3 A ( 3xz k r, 3yz 5 r, 5 r 3 ) [ 3 r 5 (r z ) k m ] 3( m r) + r r3 r 5 B ( A) ( f) ( y ( z f) z ( y f), z ( x f) x ( z f), x ( y f) y ( x f)) 6. ( A) x ( y A z z A y ) + y ( z A x x A z ) + z ( x A y y A x )

104 ( ( A) ) x y ( x A y y A x ) z ( z A x x A z ) y x A y + z x A z ( y y + z z )A x x ( x A x + y A y + z A z ) ( x x + y y + z z )A x x ( A) A x ( ( A) ) ( ( A) ) y z y ( A) A y z ( A) A z ( A ( A) A

105 ?? n m ( n m ), 3 ( ) ( ) A A A n v A v + A v + + A n v n A A A n v A v + A v + + A n v n A k A k A kn v n A k v + A k v + + A kn v n A v n (A v) j A ji v i i n m n m s s 6. A A A A n ka ka ka n A A A n ka ka ka n ka k A k A k A kn ka k ka k ka kn (ka) ij ka ij n m A ± B A A A n A A A n A m A m A mn ± B B B n B B B n B m B m B mn

106 6 A ± B A ± B A n ± B n A ± B A ± B A n ± B n A m ± B m A m ± B m A mn ± B mn (A ± B) ij A ij ± B ij n m m k n m A m k B AB m (AB) ij A ip B pj m k m k m n m B B B k B s B B B k B. ( ) b b b..... k, B s bs. B m B m B mk B ms p AB (A b A b A b k ) 3 3 AB BA

107 (a) ( 3 ) ( 3 5 ) ( 7 3 ) (b) ( ). (x, ct) (c ) x v (x, ct ) ( x ct ) β β β β β β ( x ct ) (β v c ) (a) x v t x x, x x, x (b) l x x x x l β (c) v v (x, ct ) (x, ct) β, β v /c (d)

108 (a) (b) ( 3 ) ( 3 5 ) ( 7 3 ( ) ) ( 3 ) ( 4 6 ) (,, ) ). (a) x x βct β, x x βct β (b) l x x x x l β β [(x βct) (x βct)] x x β ( ) (c) β v /c ( x ct ) (d) ( x ct ) β ( ) β β x β ct β β β β β β β β β β β β β β ( + ββ β β ( β )( β ) β β + ββ ( x ct ) ) ( x ct ( + ββ β β ) ( ) x ( β )( β ) β β + ββ ct + ββ (β+β ) ( ) (+ββ ) x ( β )( β ) (β+β ) ct (+ββ ) )

109 6 3 (β + β )/( + ββ ) ω ω + ββ ( + ββ ) (β + β ) + ββ β β β β + ββ ( β )( β ) cω cω c(β + β ) ( + ββ ) v + v + (vv /c ) < c

110 A ij i j I I I ij δ ij δ ij δ ij { (i j ) ( ) n n n n A A AI IA A n (AI) ij A ik δ kj A ij, (IA) ij k n δ ik A kj A ij k A B I B A A AA A A I ( a b A c d ) ad bc A ( d b (ad bc) c a ) ad bc 3 3 x, x,..., x n n a x + a x + a n x n b a x + a x + a n x n b. a n x + a n x + a nn x n b n

111 7 5 a a a n x b a a a n x A., x, b b a n a n a nn x n b n A x b A A A A x A b x A b x 7. A A (deta ) ( A ) ( ) a b A A ad bc c d A 3 3 A A A 3 A A A A 3 A 3 A 3 A 33 n n A A A A 33 + A A 3 A 3 + A 3 A A 3 A 3 A A 3 A A A 33 A A 3 A 3 A P (sgnp )A p A p... A npn (p p... p n ) (... n) { : (... n) (p p... p n ) sgnp : (... n) (p p... p n ) ( 3) ( 3), ( 3), ( 3 ), (3 ), ( 3 ), (3 ) 6 ( 3) ( 3) ( 3) ( 3) ( 3 ) ( 3) (3 ) ( 3) ( 3) ( 3 ) ( 3) (3 ) (3 )

112 7 6 (p p... p n ) (... n) a (a x, a y, a z ) b (b x, b y, b z ) a b e x e y e z a x a y a z b x b y b z 7.3 A x ( x ) A x λ x (λ ) λ A x A A λ A x λ x x A x λ x (A λi) x (I ) (A λi) (A λi) x (A λi) x A λi A n n n n ( ) ) A A λi ( ) ( λ λ λ ± ± x ( ) ( α β ) ( α β λ ) ( α β ) ) λ λ β λα, α βλ ( λ) N ( ), N ( )

113 7 7 N, N ( ) N N / ( )

114 (a) (d) cos θ sin θ r sin θ r cos θ (b) sin θ sin ϕ sin θ cos ϕ cos θ r cos θ sin ϕ r cos θ cos ϕ r sin θ r sin θ cos ϕ r sin θ sin ϕ (c) (e) x y z x y z. A, B AB A B (a) AB A B (b) cos θ sin θ sin θ cos θ 3. (c) ( i i ) cos θ sin θ sin θ cos θ cos θ 3 sin θ 3 sin θ 3 cos θ 3 (i i ) 4. CO M m k x, x, x 3 m d dt x k(x x ), M d dt x k{(x 3 x ) (x x )}, m d dt x 3 k(x 3 x ) O C O x i A i sin(ωt) (i,, 3) ω

115 (a) cos θ sin θ r sin θ r cos θ cos θ(r cos θ) ( r sin θ) sin θ r cos θ + r sin θ r (b) (c) (d) sin θ sin ϕ sin θ cos ϕ cos θ r cos θ sin ϕ r cos θ cos ϕ r sin θ r sin θ cos ϕ r sin θ sin ϕ + + r sin 3 θ cos ϕ r cos θ sin θ sin ϕ r cos θ sin θ cos ϕ r sin 3 θ sin ϕ r sin 3 θ r cos θ sin θ r sin θ (e) x y z x y z yz + zx + xy yx xz zy (z y)x (z y )x + yz zy (z y) [ x (z + y)x + yz ] (z y)(x y)(x z) (x y)(y z)(z x) ( a b. (a) A c d ) ( p q, B r s ) A ab cd, B ps qr, ap + br aq + bs AB cp + dr cq ds (ap + br)(cq + ds) (aq + bs)(cp + dr) adps + bcrq adqr bcsp (ad bc)ps (ad bc)qr (ad bc)(ps qr) A B

116 7 (b) ( ) cos θ sin θ cos θ sin θ sin θ sin θ cos θ cos θ cos θ sin θ sin θ cos θ cos θ 3 sin θ 3 sin θ 3 cos θ 3 cos θ cos θ 3 sin θ sin θ sin θ 3 sin θ cos θ θ sin θ 3 + sin θ sin θ cos θ 3 sin θ cos θ 3 + cos θ sin θ sin θ 3 cos θ cos θ sin θ sin θ 3 cos θ sin θ cos θ 3 cos θ sin θ 3 sin θ cos θ cos θ 3 (c) 3. λ i i λ λ λ ± ( i i ) ( α β ) ( α λ β ) ) iβ λα, iα λβ ) λ N ( i λ N ( i (N, N ) 4. m d dt x k(x x ), x i A i sin(ωt) (i,, 3) M d dt x k{(x 3 x ) (x x )}, m d dt x 3 k(x 3 x ) ω k m k M m( ω )A sin(ωt) k(a A ) sin(ωt) M( ω )A sin(ωt) k(a 3 A + A ) sin(ωt) m( ω )A 3 sin(ωt) k(a A 3 ) sin(ωt) k m k M ω k M k m ω k m A A A 3 ω k k m m k ω k k M M M (ω k k ω k m ) (ω k M ) k Mm (ω k m ) m m ω (ω k m )(ω k m k M ) k k ω, ± m, ± m + k M

117 8 8. A A T A t A A A n A A A n A A A n A. A T A A A n A k A k A kn A n A n A kn (A T ) ij A ji, ( ) a, b b a b a T b n b (a a... a n ) a. b + a b b + + a n b n a k b k k b n a R a, b R b a T a T R T a b a T R T R b a T b R T R I 8. ( a, b) ( a ) T n b a kb k a b U k ( a ) T b (U a) T (U b) ( a ) T (U ) T U b ( a ) T b, (U ) T U I A (A ) T A A (A ) T (A ) ij A ji.

118 8 U U UU I H H H λ v H v λ v ( v ) T ( v ) T H v λ( v ) T v vi H ij v j λ i,j i v i v i. H ( i,j v i H ij v j ) i,j v i H ijv j i,j v i H ji v j i,j v j H ji v i i,j v i H i j v j i v i v i λ H λ λ v v H v λ v, H v λ v ( v ) T H λ ( v ) T ( v ) T H λ ( v ) T (H H ) ( v ) T ( v ) T H v λ ( v ) T v ( ) λ ( v ) T v λ ( v ) T v λ ( v ) T v (λ λ )( v ) T v λ λ ( v ) T v v v 8.3 H λ, λ,... λ n x, x,... x n ( ) ( ) U ( x x... x n ) : x x x x x x x n U x x x x x x x n U ( x. x... x n ). I..... x n x n x x n x x n x n

119 8 3 U HU x x. H( x x... x n ) U HU x x. (λ x λ x... λ n x n ) x n λ x x λ x x λ n x x n λ x x λ x x λ n x x n x n λ λ λ x n x λ x n x λ n x n x n λ n U HU H U : A T A ( ) : A T A : A A (A ) ij (A ij )

120 ( cos θ sin θ. R sin θ cos θ ). σ ( ), σ ( i i ), σ 3 ( ) σ, σ, σ 3 3. x, y, z U(x, y, z) U U( U )U 4. (TrA i (A) ii ) a) Tr(A + B) TrA + TrB b) Tr(AB) Tr(BA) c) Tr(B AB) Tr(A) 5. X e X n n! Xn θ e iθσ sin θ (a) (iθσ ) n (n + )! ( )n θ n+, cos θ n (n)! ( )n θ n (b) (iθσ ) n n (n m + ) (n m) (c) n! (iθσ ) n sin θ, cos θ n 6. H H U ( ) m ik H (m, k ) ik m

121 R T R ( cos θ ) ( sin θ cos θ sin θ ) sin θ cos θ sin θ cos θ ( cos θ + sin θ cos θ sin θ + sin θ cos θ sin θ cos θ + cos θ sin θ ( sin θ) + cos θ ) ( ) I. σ σ ( σ σ σ, σ 3 ) ( ( ) ( ) ) ( ( 3. U UU I U (UU ) I O ( U)U + U( U ) O ( U)U U + U( U )U OU O U + U( U )U O ) ) σ I U U( U )U 4. (a) Tr(A + B) i (A + B) ii i (A ii + B ii ) i A ii + i B ii TrA + TrB (b) Tr(AB) i k (AB) ii A ik B ki i k k (BA) kk Tr(BA) B ki A ik i (c) b) Tr(B AB) Tr ( B (AB) ) Tr ( (AB)B ) Tr(ABB ) TrA

122 (a) (iθσ ) θ σ θ I (b) n m + (m ) (iθσ ) m+ i m+ θ m+ σ m+ i( ) m θ m+ σ n m (iθσ ) m i m θ m σ m ( ) m θ m I (c) [ ] e iθσ n n! (iθσ ) n m (m)! (iθσ ) m + (m + )! (iθσ ) m+ [ ] m (m)! ( )m θ m I + i (m + )! ( )m θ m+ σ ( ) cos θ i sin θ cos θi + i sin θσ i sin θ cos θ 6. m λ ik ik m λ λ m ±k λ m ± k (λ m) k m k : ( i ), m + k : ( i ), U ( i i ) (U H ) U HU ( i i ( i ) ( ) ( ) m ik i ik m i ) ( ) m k im + ik i ik + im k + m ( m k k + m im + ik ik im im + ik ik + im m + k + k + m ( ) m k m + k )

123 III A ( )

124 x a (a > ) i ( ) x ±i a x, y z x + iy (a n x n + a n x n + a x + a ) ( ) ( ) ) m d x(t) dt kx(t) x(t) Ae iωt (A, ω ) k (iω) Ae iωt kae iωt ω ± m ( ) x(t) A e i( k/m)t + A e i( k/m)t 9. z x + i y (x, y ) (real part) : Re(z) x (imaginary part) : Im(z) y : z x i y ( z ) : z x + y z z z z z z, z z, z z z + z z + z z

125 9 9 : Re(z) Im(z) z x + iy z x + iy x ( θ) arg[z] θ z z z (cos θ + i sin θ) Im(z) y z θ O zx+iy x z Re(z) z z z (cos θ + i sin θ ) z (cos θ + i sin θ ) z z [(cos θ cos θ sin θ sin θ ) + i (sin θ cos θ + cos θ sin θ )] z z [cos(θ + θ ) + i sin(θ + θ )] 9.3 θ iθ e iθ cos θ + i sin θ z r z re iθ, z re iθ cos θ sin θ cos θ + i sin θ m ( ) m (m)! θm + i m ( ) m (m + )! θm+ k k! (iθ)k e iθ 9.4 x + y + z (,, ) xy C (a, b, ) P ( a P a + b +, b a + b +, a + b ) a + b + z O P (α,β,γ) b y C P x a C

126 9 9.5 f(z) z z ( g(z, z ) z + 5z ) (h(z, z,...)) 9.5. : c, c,... f(z) c n z n + c n z n + + c z + c : a, a,... b, b,... : z x + iy (x, y ) f(z) a mz m + a m z m + + a z + a b n z n + b n z n + + b z + b e z e x+iy e x e iy e x (cos y + i sin y) : z re iθ (r, θ ) (ln x log e x) ln z ln[re iθ ] ln[r] + ln[e iθ ] ln r + iθ sin z i [eiz e iz ], cos z [eiz + e iz ], tan z sin z cos z sinh z [ez e z ], cosh z [ez + e z ], tanh z sinh z cosh z ([ ]sinh : sin, cos h sinh cosh ) (e z+w e z e w, (e z ) w e zw, ln(zw) ln z+ln w ) 9.5. f(z) z / z re iθ re iθ e nπi re i(θ+nπ) (n ) { r (n ) z / r (n ) z z re iθ θ < π π < θ π

127 9 ( ) [ : θ < π ].. i i (e iπ/ ) i e π/ 3. ln( + 3i) ln ( ) i (e iπ e ln ) i e π e i ln e π [cos(ln ) + i sin(ln )] ( ).43 [cos(.693) + i sin(.693)] ln + ln e iπ/3 ln + π 3 i [.43 ( i) i ( )] ( 3 + i ln + ln cos π 3 + i sin π ) 3 4. sin(i) i (eii e ii ) i (e e) 5. cosh( π 4 i) (eiπ/4 + e iπ/4 ) cos π f(z) z z ( ) : f(z) z (z )(z + ) z f() z (z ) f(z) z + (z ), f() z : f(z) z + z (z )f(z) z z f(z) z a (z a) n f(z) (n ) z a z a n : ) f(z) e z z

128 z x + iy re iθ (x, y, r z, θ θ < π) x y r, θ (a) z + + i (b) ( + i 3)z (c) z (d) (z 3 ) (e) /z (f) /z (g) (z + z )/ (h) (z z )/ (i) z (j) z/ z. x + iy (a) e iπ/6 (b) e iπ/4 (c) e iπ/3 (d) e iπ/ (e) e iπ/4 (f) e iπ/ (g) e i3π/ (h) e i3π/4 (i) e i7π/4 (j) e i6π/3 3. z 3 z x + iy 4. (cos θ + i sin θ) n cos(nθ) + i sin(nθ) (n ) (hint: (e iθ ) n e inθ ) 5. x R I (hint: R + ii ) 6. n cos(kx) cos x + cos(x) cos(nx), k n sin(kx) sin x + sin(x) sin(nx) k d dt f(t) ω f(t) (ω ) f(t) Ce at (C, a ) a f(t) 7. z, z z + z z + z 8. z (a) (b) (c) z + < < z < < arg[z] < π/4 z + z + < 6 9. π < arg[z] π ( ) 3 ( + i 3 + i a) (i) 4 b) i c) ( i) i d) e). π < arg[z] π a) ln b) ln i c) ln( + i) ( ) 3 [ ] d) ln + i ( + i 3) e) ln ( 3 + i) ) 3i

129 9 3. π < arg[z] π a) cos(i) b) tan( i) c) sin(i ln ) d) cos(π + i) e) sec( i) cos( i). π < arg[z] π a) sinh() b) sinh(iπ) c) cosh(i π 6 ) d) tanh(i π ) e) sinh( + iπ) 3 3. e z e w e (z+w) (z, w ) 4. ln(zw) ln z + ln w (z, w ) 5. sin(z + w) sin z cos w + cos z sin w (z, w ) 6. cosh(z + w) cosh z cosh w + sinh z sinh w (z, w )

130 x, y a) (x + ) + i(y + ) (r cos θ + ) + (r sin θ + ) arctan( r sin θ + r cos θ + ) b) (x 3y) + i( 3x + y) r θ + π 3 c) (x y ) + i(xy) r θ d) x 3 3xy + i(3x y y 3 ) r 3 3θ x e) (x + y ) i y θ x + y r x f) (x + y ) + i y θ x + y r g) x r cos θ (x > ), π (x < ) h) iy r sin θ π (y > ), 3π (y < ) i) x + y r j) x x + y + i y x + y θ. a) e) i) 3 + i b) + i c) 3 + i d) i i f) i g) i h) + i i j) 3. z, e π 3 i, e π 3 i, i, i 4. (cos θ + i sin θ) n cos(nθ) + i sin(nθ) n n k (k ) (cos θ+i sin θ) k cos(kθ)+i sin(kθ) n k+ (cos θ + i sin θ) k+ (cos θ + i sin θ) k (cos θ + i sin θ) (cos(kθ) + i sin(kθ))(cos θ + i sin θ) cos(kθ) cos θ sin(kθ) sin θ + i [sin(kθ) cos θ + cos(kθ) sin θ)] ( ) cos[(k + )θ] + i sin[(k + )θ] n k + n

131 R + ii n n [cos(kx) + i sin(kx)] e ikx e ix + e ix + + e inx k k ( ) eix ( e inx ) eix e inx/ (e inx/ e inx/ ) e ix e ix/ (e ix/ e ix/ ) [cos( R cos( (n + )x (n + )x ) + i sin( )] sin(nx/) sin(x/) (n + )x ) sin(nx/) sin(x/) i(n+)x/ ( i) sin(nx/) e ( i) sin(x/) + )x, I sin((n ) sin(nx/) sin(x/) 6. d dt f(t) ω f(t) f(t) Ce at a Ce at ω Ce at a ω a ±iω f(t) C e iωt + C e iωt 7. z x + iy, z x + iy (x,, y, ) z x + y, z x + y, z + z (x + x ) + (y + y ) ( z + z ) z + z ( x + y + x + y) (x + x ) + (y + y ) x + y + x + y + (x + y)(x + y) (x + x + x x + y + y + y y ) [ ] (x + y)(x + y) (x x + y y ) [ ] (x + y)(x + y) (x x + y y ) (x + y )(x + y ) (x x + y y ) x x + y x + x y + y y (x x + x x y y + y y ) y x + x y x x y y (x y x y ) ( z + z ) z + z z + z, z + z z + z z + z

132 9 6 z, z z, z, z z ( ) z, z, z z Im(z) z z z z z + z z + z O Re{z} 8. (a) (b) (c) Im z Im z Im z 5 - O Re z - - O - π/4 Re z -3 O 3 Re z (x + ) + y < (x/3) + y /5 < 9. (a) (i) 4 4 i 4 6( ) 6 (b) i (e ) i e (c) ( i) i (e iπ/ ) i e π/ ( ) 3 + i (d) (e iπ/4 ) 3 e i3π/4 + i. (e) ( ) 3i 3 + i (e iπ/6 ) 3i e π/ (a) ln (b) ln i ln e iπ/ π i [ ] ( + i) (c) ln( + i) ln ln( ) + ln e iπ/4 ln + π 4 i ( ) 3 (d) ln + i ln e i3π/4 3π 4 i [ ] [ ] [ ] ( + i 3) ( + i 3) (e) ln ( e ln 3 + i) ( iπ/3 ln ln(e iπ/6 ) π 3 + i) e iπ/6 6 i

133 9 7.. (a) cos i (ei i + e ii ) (e + e) (b) tan( i) sin( i) cos( i) (e i( i) e i( i) ) i (e i( i) + e i( i) ) (e e ) i (e + e ) ) i(e (e + ) (c) sin(i ln ) (eii ln e ii ln ) i (d) cos(π + i) (ei(π+i) + e i(π+i) ) (e) sec( i) (e ln e ln ) i e e ( ) i (e + e) cos( i) e i( i) + e i( i) e + e (a) sinh() e e e4 e (b) sinh(iπ) eiπ e iπ ( ) ( ) (c) cosh(i π 6 ) ei π 6 + e i π 6 cos π 3 6 (d) tanh(i π 3 ) ei π 3 e i π 3 i sin(π/3) e i π 3 + e i π 3 cos(π/3) 3i (e) sinh( + iπ) e +iπ e ( +iπ) 3 4 i e e 4 + (e e ) e4 e 3. z x + iy, w x + iy (x,, y, ) e z e w e x +iy e x +iy e x (cos y + i sin y )e x (cos y + i sin y ) e x +x [(cos y cos y sin y sin y ) + i(sin y cos y + cos y sin y )] ( ) e x +x [cos(y + y ) + i sin(y + y )] e (x +x )+i(y +y ) e z+w 4. z r e iθ, w r e iθ (r,, θ, ) ln(zw) ln(r e iθ r e iθ ) ln(r r e i(θ +θ ) ) ln(r r ) + i(θ + θ ) ln(r ) + ln(r ) + i(θ + θ ) ln(r ) + iθ + ln(r ) + iθ ln z + ln w 5. 3 e z +z e z e z sin(z + w) i (ei(z+w) e i(z+w) ) [ (e iz e iz )(e iw + e iw ) + (e iz + e iz )(e iw e iw ) ] 4i (eiz e iz ) (e iw + e iw ) + (eiz + e iz ) (e iw e iw ) sin z cos w + cos z sin w i i 6. cosh(z + w) (ez+w + e (z+w) ) [ (e z + e z )(e w + e w ) + (e z e z )(e w e w ) ] 4 (ez + e z ) (e w + e w ) + (ez e z ) (e w e w ) cosh z cosh w + sinh z sinh w

134 8. z z lim z z f(z) f(z) Im(z) z z Re(z) O f(z) z z g(z, z ) z z z reiθ z r lim g(z, z ) lim z re iθ r re iθ e iθ θ. df dz lim f(z + z) f(z) z z z f(z) z f(z) z f(z) z f(z) z x+iy Re[f(z)] u(x, y) Im[f(z)] v(x, y) f(z + z) u(x + x, y + y) + iv(x + x, y + y), u(x + x, y + y) u(x, y) + x u x + y u y + O( x y, x, y ), v(x + x, y + y) v(x, y) + x v x + y v y + O( x y, x, y ), ( f(z + z) f(z) x u + i ) ( x v x + y u + i ) y v y.

135 9 f(z + z) f(z) z ( x u + i ) x v ( x x + i y + y u + i ) y v y x df ( dz x y df dz i x u + i ) x v ( y u + i ) y v y x + i y. x u y v, y u x v ( )

136 3.3. a) lim n ( + i) n b) lim z e iπ/3 (z + z + ) c) lim z i (z 3 + i) (z i) d) lim z e z z. a) z + z + b) sin z c) z z z d) i 3. f(z) u(x, y) + iv(x, y) z re iθ r, θ ( u r, u θ, v r, v θ ) 4. (a) z ( ) (b) e z ( ) (c) ln z ( ) (d) cos z (cos z (b) ) (e) arctan z ( w arctan z z tan w (e iw e iw )/i(e iw + e iw ). e iw (b), (c) ) 5. f(z) f(z) f(z) 6. z x+iy (x, y ) f(z) u(x, y)+iv(x, y) (u Re[f], v Im[f]) [ ] x + y u(x, y), [ ] x + y v(x, y)

137 3.4. (a) lim n ( + i) n lim n (b) [ e iπ/4 ] n lim (z + z + ) e iπ/3 + e iπ/3 + z e iπ/3 + (z 3 + i) (c) lim z i (z i) lim z i (z 3 i 3 ) (z i) (d) z re iθ (r, θ ) (e z ) lim z z lim r e r(cos θ+i sin θ) re iθ. z x + iy (x, y ) lim z i (z i)(z + iz ) (z i) lim r 3 i i + + 3i i + ii 3 [ e r cos θ ] {cos(r sin θ) + i sin(r sin θ)} re iθ [ lim [ + r cos θ + O(r )][ + O(r ) + i{r sin θ + O(r )}] ] r re iθ [ lim r(cos θ + i sin θ) + O(r ) ] eiθ r re iθ e iθ (a) z +z + (z +) (x++iy) (x+) y +i(x+)y u (x+) y, v (x + )y (b) u (x + ), x y u y, x u y v, x v y, x u + i v (x + ) + yi (z + ) x v (x + ) y y u x v sin z eiz e iz i i [ei(x+iy) e i(x+iy) ] i [e y {cos x + i sin x} e y {cos x i sin x}] sin x (ey + e y ) + i cos x (ey e y ) u + iv x u cos + e y ) x(ey, x v sin e y ) x(ey, y u sin e y ) x(ey, y v cos + e y ) x(ey x u + i x v cos + e y ) x(ey + i( sin x) (ey e y ) [ey (cos x i sin x) + e y (cos x + i sin x)] [ey e ix + e y e ix ] [e i(x+iy)+ei(x+iy ] [eiz + e iz ] cos z

138 3 (c) z x + y u + iv u x + y, v x u x, y u y, x v, y v (d) z z i y u + iv u y, v x u, y u, x v, y v 3. z x + iy re iθ r cos θ + ir sin θ x r cos θ, y r sin θ u r v r u θ v θ x u r v x r x θ x θ x + y u r y x + y v r y u x + y u θ y v x + y v θ y u u cos θ + sin θ x y v v cos θ + sin θ x y u u r sin θ + r cos θ x y v v r sin θ + r cos θ x y r u r v θ r v r + u θ r cos θ( u x v y r cos θ( u y + v x ) + r sin θ(u y + v x ) ) + r sin θ(u x v y ) r u r v θ, r v r u θ 4. (a) d (z + w) z dz z lim w w lim w zw + w w z (b) e z ex + iy e x (cos y + i sin y) u + iv u e x cos y, v e x sin y d dz ez u x + iv x ex cos y + ie x sin y e x ((cos y + i sin y) e z (c) ln z ln re iθ ln r u + iv u ln r, v θ d u ln z dz x + iv x ln r x + i θ x

139 33 r x + y ln r x ln(x + y ) x x y r sin θ x x + y y x r sin θ sin θ + r x x x θ sin θ + r cos θ x + y x xy r + x θ x θ x y r y x + y d ln z dz x x + y i y x + y x iy x + y x + iy z (d) d dz cos z d dz (eiz + e iz ) (ieiz ie iz ) i (eiz e iz ) sin z (e) w arctan z z tan w (eiw e iw ) i(e iw + e iw ) eiw t z (t (/t)) i(t + (/t)) t i(t + ) i(t + )z t t (iz ) (iz + ) t e iw + iz iz ( ) iw ln( + iz) ln( iz) ( z ) i dw i dz dw dz i + iz ( i) iz [ i( iz) + + z ] i( + iz) + z + z 5. f(z) f(z) u v x u y v, y u x v. f(z) v x u, y u. u x, y f(z) u

140 34 6. x u y v, y u x v. x x u x y v y x v y y u y u y y v y x u x y u x x v x v [ ] x + y u(x, y), [ ] x + y v(x, y)

141 35. b a f(x)dx a b z z f(z) /z z z i Im z C z - O - C C3 Re z. C : z + ( + i)t ( t ) dz ( + i)dt C z dz dz(t) dt ( + i) dt z(t) dt [ + ( + i)t] ln[ + ( + i)t] ln i ln π i. C : z e iθ ( θ π/) dz ie iθ dθ π/ C z dz dz(θ) z(θ) dθ dθ 3. C : z e iϕ ( ϕ 3π/) dz ie iϕ dϕ 3π/ C 3 z dz dz(ϕ) 3π/ z(ϕ) dϕ dϕ π/ e iθ ieiθ dθ π i e iϕ ( i)e iϕ dϕ 3π i (+ ) : f(z)dz c

142 36. D ( ) f(z) D C f(z)dz [ ] z x + iy f(z) u v C f(z)dz C C f(z) u(x, y) + iv(x, y) (u + iv)(dx + idy) [(udx vdy) + i(vdx + udy)] C ( ) [( S u ydxdy ) ( v xdxdy + i v ydxdy + )] u xdxdy [ ( S u y + ) ( v x i v y )] u x dxdy ( ) f(z) f(z)dz C +( C ) f(z)dz C f(z)dz C z C z z C f(z)dz f(z)dz C C sin(x )dx cos(x )dx e iz (C C + C + C 3 ) e iz C e iz dz C : z x (x R), dz dx C e iz C : z Re iθ (θ π/4), dz ire iθ dθ dz + C e iz dz + Im z O C 3 e iz C 3 : z te iπ/4 (t R ), dz e iπ/4 dt [ R π/4 ] lim e ix dx + e ir e iθ ire iθ dθ + e it e iπ/ e iπ/4 dt R R π/4 cos(x )dx + i sin(x )dx + lim e ir cos θ e R sin θ ire iθ dθ ( + i) R C 3 π/4 C dz C R z 8 Re z e t dt

143 37 R sin(x )dx cos(x )dx e t dt π.3 f(z) < z z < r z z < r z C Res[f] zz f(z)dz πi C ) f(z) (z z ) n C z z + Re it (t π) Res[f] zz πi πi C π (z z ) n dz πi ir n+ e i(n+)t dt π (Re it ) n ire it dt { (n ) (n ) C f(z) ( ) z, z,... z n n f(z)dz πi Res[f] zzk C k z z f(z) k < z z < R f(z) f(z) f(z) a k (z z ) k + a (k ) (z z ) k + + a (z z ) + a + g(z). g(z) z z < R Res[f] zz πi πi f(z)dz [ a k (z z ) + + a k (z z ) + a + g(z) ( ) a πi (z z ) dz a (k )! lim z z ] dz d k dz k [(z z ) k f(z)]

144 38 ) x + a dx (a ) x a tan θ z + a dz C +C C +C Im z z (z ai)(z + ai) dz a C R C Re z -R O R ) (C : z x (x R R)) (C : z Re iθ (θ π)) R R C z + a dz R x + a dx x + a dx π lim R π C z + a dz C +C z + a dz a cos θ + a C +C [ ] R e iθ + a ireiθ dθ x + a dx lim R z + a dz C +C (z ai)(z + ai) dz πires πi lim (z ai) z + a zai z ai (z ai)(z + ai) π a x + a dx π a dx x 4 + a4 e iθ z cos θ dθ (a a < ) ( z + z ) dz izdθ C π C C C a cos θ + a dθ dz [ a(z + ) + z a ] iz i [az ( + a )z + a] dz i a(z a)(z /a) dz i πi Res[ a(z a)(z /a) ] za i πi z a lim(z a) a(z a)(z /a) π a Im z C - O a - z Re z

145 39.4. I C zdz (a) z + i (b) z z + i (c) z i z + i. C I z n dz (a) n (b) n (c) n C 3. I + e ikx dx (k > ) (a) R z R z R (C ) R (C ) e ikz Im[z] R C (b) (C ) z Re iθ R -R C R Re[z] (c) I 4. f(z) e z R e x cos(ax)dx π e a Im(z) a -R O R Re(z) 5. (x + )(x + 9) dx 6. x dx ( < θ < π) ( + x )( x cos θ + x )

146 4 7. p p < π + p cos θ dθ 8. a πi lim e iax ϵ + (x iϵ) dx 9. I + ii I I n I π e cos θ cos(nθ sin θ)dθ, I π e cos θ sin(nθ sin θ)dθ. cos mx dx (a, m > ), x + a

147 4.5. (a) C : z ( + i)t (t ) dz ( + i)dt C zdz ( + i)t( + i)dt ( + i) tdt ( + i ) i (b) C C + C, C : z t (t ) C : z + is (s ) C dz dt C dz ids C zdz zdz + zdz C C tdt + ( + is)ids + i i (c) C C + C, C : z it (t ) C : z s + i (s ) C dz idt C dz ds C zdz zdz + C zdz C itidt +. C : z e iθ (θ π) dz ie iθ dθ (a) n C z n dz i π π π (e iθ ) n ie iθ dθ i (s + i)ds + + i i e i(n+)θ dθ (cos[(n + )θ] + i sin[(n + )θ]) dθ 3. (b) n C z n dz π e iθ ie iθ dθ (c) n n m m C z n dz i π π π π (e iθ ) m ie iθ dθ i idθ πi e i(m )θ dθ (cos[(m )θ] i sin[(m )θ]) dθ (a) e ikz z (b) C : z Re iθ (θ π) dz ire iθ dθ C e ikz dz π π e ikreiθ ire iθ dθ π ire ikr cos θ e kr sin θ e iθ dθ e ikr(cos θ+i sin θ) ire iθ dθ e ikr cos θ e iθ k > < θ < π kr sin θ < R Re kr sin θ

148 4 (c) C z x (x ) dz dx e ikz dz I (a), (b) C e ikz dz C e ikz dz + C e ikz dz I + C I 4. e z z dz C x R R dx C R + it a idt C 3 p + ia R R dp C 4 R + is a ids z e z dz C +C +C 3 +C 4 e z C e z C e z C 3 C 4 e z dz dz dz R R a R R e x dx π (R ) a e (R+it) idt ie R e irt e t dt (R ) e (p+ia) dp R R e p e iap e a dp R e a e p [cos(ap) i sin(ap)] dp R R e a e p cos(ap)dp dz ( ) R e a a (e p sin(ap) ) e x cos(ax)dx (R p x ) a e (R+is) ids ie R e irs e s ds (R ) π + e a e x cos(ax)dx e x cos(ax)dx + e x cos(ax)dx π e a 5. 3 (R ) z ±i, ±3i (z + )(z + 9) z i, 3i [ ] [ ] dz πires + πires C +C (z + )(z + 9) (z + )(z + 9) (z zi + )(z + 9) z3i πi lim(z i) + πi lim z i (z + )(z + 9) (z 3i) z 3i (z + )(z + 9) [ ] [ πi i( + 9) + π ( 9 + )6i 8 4] π

149 43 (z + )(z + 9) dz C π C (z + )(z + 9) dz (x + )(x + 9) dx (R e iθ + )(R e iθ + 9) ireiθ dθ (R ) (x + )(x + 9) dx π 6. I x ( + x )( x cos θ + x ) dx x (x i)(x + i)(x e iθ )(x e iθ ) dx z ±i, e iθ, e iθ R z i, e iθ I πires[f] zi + πires[f] ze iθ (f ) -R - R x O x x θ x z R Res[f] zi Res[f] ze iθ i(i e iθ )(i e iθ ) i[ i(e iθ + e iθ ) + ] e iθ (e iθ + i)(e iθ i)(e iθ e iθ ) e iθ (e iθ + )i sin θ e iθ (e iθ + e iθ )i sin θ cos θ + i sin θ 4i cos θ sin θ 4i sin θ + 4 cos θ [ I πi 4 cos θ + 4i sin θ + ] π 4 cos θ sin θ i( i) cos θ 7. ) π π + p cos θ dθ ip + (p/)(e iθ + e iθ ) C dθ (/p)z + z + dz C dz + (p/)(z + ) iz z z + (/p)z + p [ ± p ] α p [ + p ], β p [ p ] β > f(x) x + (/p)x + (x )

150 44 f()f( ) ( + p )( p ) 4( ) < f(x) < x < p α < C α C dz (/p)z + z + [ ] πires (/p)z + z + πi (α β) πi (/p) p zα πi lim z α (z α) (z α)(z β) π + p cos θ dθ ip πi (/p) p π p 8. f(z) eiaz z iϵ z iϵ a > : f(z) C f(z)dz e iax π iar(cos θ+i sin θ) e dx + lim ire iθ dθ R Re iθ iϵ x iϵ e iax x iϵ dx + πires[f] ziϵ πi lim z iϵ e iaz πie ϵa (ϵ ) a < : f(z) C f(z)dz e iax π iar(cos θ+i sin θ) e dx + lim ire iθ dθ R Re iθ iϵ x iϵ e iax x iϵ dx + (f(z) ) 8 8 πi lim e iax { (a > ) ϵ + x iϵ dx (a < ) 8 ε x O ε x O 8 z 8 z 8

151 45 9. I + ii π π e cos θ [cos(nθ sin θ) + i sin(nθ sin θ)]dθ e cos θ e i(nθ sin θ) dθ π e inθ e e iθ dθ z e iθ dz ie iθ dθ C I + ii C z n ez i z dz i C π n! lim z ez π n! I π/n! I e z ( ) n [ ] d n+ ez dz iπi lim z zn+ n! z dz z n+. 3 (R ) eimz z ±ai (z + a ) z ai C+C e imz C e imz z + a dz C e imz dz πires z + a infty π dz lim z + a R lim R π e imx [ x + a dx e imreiθ e imz ] (z + a ) zai R e iθ + a ireiθ dθ lim R e imr cos θ mr sin θ e πi e ma ai π a e ma cos(mx) + i sin(mx) dx x + a π R e iθ + a ire iθ dθ cos(mx) x + a imr(cos θ+i sin θ) e ire iθ dθ R e iθ + a cos(mx) x + a π a e ma

152 46. z z R f(z) C : z z R πi C f(w) w z dw [ πi lim (w z) f(w) ] f(z) πi w z w z x z x z R w C z f (z) πi f (z) πi C C f(w) (w z) dw f(w) (w z) dw 3. f (n) (z) n! πi C f(w) dw (w z) n+. z z < R z C w w z R > z z w z (w z ) (z z ) (w z ) f(z) f(w) ( ) z z n dw πi C w z n w z n n! f (n) (z )(z z ) n n [ z z w z ] (w z ) [ πi C ( ) ( ) z z n n w z ] f(w) dw (z z (w z ) n+ ) n

153 47.3 < R < z z < R f(z) f(z) πi C +( C )+C 3 +C 4 C w w z > z z f(w) w z dw πi C f(w) w z dw πi C f( w) w z d w w z (w z ) z z w z (w z ) ( ) z z n n w z C w w z < z z w z (z z ) w z z z (z z ) ( ) w n z n z z C C3 C4 z x C z x R R f(z) [ ] f(w) dw (z z πi n C (w z ) n+ ) n + πi a n (z z ) n, n a n πi C (w z ) (n+) f(w)dw. [ ] f( w) ( w z ) k dw C k (z z ) k ) sin z ( z ) z 3 z 3 3! z3 + n ( ) n (n + )! zn.4 sin x sin x a b c x sin x c x < x < π/ a

154 48 y x sin x - O sin x x - z x + iy sin z sin z i [eiz e iz ] i [ei(x+iy) e i(x+iy) ] i [ e y (cos x + i sin x) e y (cos x i sin x) ]

155 49.5. a) sinh x : x x Taylor b) e z : z Taylor c) d) x e ξ dξ : x x Taylor z(z )(z ) e) cos z z : z Laurent : z Laurent f) z e /z : z Laurent. Gamma Γ(z) Re(z) > Γ(z) Γ(z + ) zγ(z) (a) < Re(z) < Γ(z) Γ( /) Γ(/) π e t t z dt Γ(z + ) z (b) z n (n,,,...) z Γ(z) z n (n,,,...) 3. f(z) n (a) f(z + z ) f(z )f(z ) n! zn. (b) z x f(x) e x f(z)

156 5.6. (a) sinh x (ex e x ) sinh x (ex e x ) [ ] x n n n! ( x)n n! n (n ) x k+ k (k + )! [ ( ) n ] x n n! (b) (c) e z e e z (z ) n e n! n x e ξ dξ x ( ξ ) n ( ) n x dξ ξ n dξ n n! n n! ( ) n x n+ n n! (n + ) (d) z(z )(z ) {(z ) + }(z ){(z ) } (z ){(z ) } (z ) [ (z ) ] [(z ) ] n (z ) n (z ) n n (e) (f) cos z z z z e /z z n n ( ) n (n)! zn n! ( z )n n ( ) n (n)! zn ( ) n z n n n!

157 5. (a) Γ(z) Γ(z + ) z z (/) Γ[ Γ( (/) + ) ] (/) Γ(/) π (b) (n + ) < Re[z] < n Γ[z] Γ(z + ) z z Γ(z + ) (z + ) Γ(z + n + ) z(z + )... (z + n) Re[z + n + ] > Γ(z + n + ) Γ(z) z,,,, n 3. (a) 4.3 (b) z x (x ) f(x) n x n n! ex

158 5 3 δ δ 3. δ δ n, m δ nm { (n m ) ( ) π e i(n m)x dx π π π (cos[(n m)x] + i sin[(n m)x])dx i cos[(n m)x] π n m (n m) π π x + ic( ) π (n m) sin[(n m)x] n m π π e i(n m)x dx δ nm π π 3. δ δ(x a) δ f(x) x a δ(x a) f(x) δ(x a)dx f(a) δ ( ) ( )/

159 3 δ δ 53 δ θ(x a) { (x a ) (x < a ) dθ/dx f(x) O a x f(x) d dx θ(x a)dx f(x)θ(x a) f( ) a f (x)θ(x a)dx f (x)dx f( ) [f( ) f(a)] f(a) x a θ(x a) δ(x a) d θ(x a) dx δ δ 3 ( r) δ 4 ( r, t) δ δ 3 ( r) δ(x)δ(y)δ(z), δ 4 ( r, t) δ(x)δ(y)δ(z)δ(t)

160 3 δ δ a) k δ 3 k b) c) k x δ(x 7) dx d) δ mk δ kn k x δ(x 4) dx. δ f(x) x a) δ(x) δ( x) b) xδ(x) c) δ(ax) a δ(x) (a ) d) xδ (x) + δ(x) 3. f(x) d δ(x a) dx f(x)δ (x a) dx f (a) δ (x a)

161 3 δ δ (a) δ k k k k 3 δ k 3 8 (b) δ kn k n δ mk δ kn δ mn k (c) x 7 y x y + 7 dx dy x δ(x 7) dx ( ) y + 7 δ(y) ( ) 7 dy 49 8 (d) x 4 (x )(x + ) x ± δ x x 4 x x + 4 x δ(x 4) dx x δ ((x )(x + )) dx x δ ( 4(x + )) dx + ( 4(x + ) y ) 8 ( ) y 4 δ(y)( + x δ ((x )4) dx (4(x ) z ) 4 )dy + 8 ( z 4 + ) δ(z) 4 dz. (a) f(x) f(x)δ(x)dx f() f(x)δ( x)dx x t dx dt f(x)δ( x)dx f( t)δ(t)( dt) f( t)δ(t)dt f( ) f() f(x) δ(x) δ( x) (b) f(x) f(x)xδ(x)dx (f(x)x) δ(x)dx f() f(x) xδ(x)

162 3 δ δ 56 (c) a > ax t adx dt (d) a > a < δ(ax) a δ(x) f(x)δ(ax)dx f(x)δ(ax)dx f( t a )δ(t) dt f() a a f( t a )δ(t) f() dt a a f( t a )δ(t) a dt f(x)[xδ (x) + δ(x)]dx [f(x)xδ(x)] (f(x)x) δ(x)dx + f() (f (x)x + f(x))δ(x)dx + f() f() + f() f(x) xδ (x) + δ(x) ( ) (b) xδ(x) x (xδ(x)) xδ (x) + δ(x) 3. f(x)δ (x a)dx [ f(x)δ(x a) ] f (x)δ(x a)dx f (a)

163 f(x) {x n } {, x, x, } f (n) (a) f(x) (x a) n n! n {sin(nx)}, {cos(nx)} π < x < π f(x) (sin, cos ) e inx f(x) C n e inx (C n x ) n C n e imx π π π π π f(x)e imx dx C n e inx e imx dx C n e i(n m)x dx π π n n C n πδ nm πc m n C n π f(x)e inx dx π π f(x) C n e inx, C n π f(x)e inx dx π π n : f(x) x ( π < x < π) C n π n C n π π π n x π n n n π π xe inx dx xe inx dx ( [ x π ( i (n) (πe inπ ( π)e inπ ) ( in) e inx] π i (n) δ n C π xdx π π i n ( )n e inx i + n n ( )n e inx [ i n ( )n e inx + i ] n ( )n e inx ( ) n+ n sin(nx) π π ( in) ) i n ( )n π π e inx dx i( ) n (e inx e inx ) n n )

164 m 3 m ( ) n+ sin(nx) ( m ) n n : δ(x) ( π < x < π) C n π δ(x)e inx dx π π π δ(x) n π einx [ ] e inx + + e inx π n n [ ] + (e inx + e inx ) [ ] + cos(nx) π n π n π + n cos(nx) π 4 3 m 3 5 π + m n cos(nx) π ( m ) f(x) π < x < π L < x < L L {e iπnx/l } f(x) n C n e iπnx/l, C n L f(x)e iπnx/l dx L L

165 < x < f(x) {e ikx } (k ) f(x) π : f(x) a (a ) C(k)e ikx dk, C(k) π f(x)e ikx dx C(k) π ae ikx dx a πδ(k) : f(x) δ(x a) C(k) π δ(x a) π δ(x a)e ikx dx π e ika e ika e ikx dk e ik(x a) dk π π 3 : f(x) e ax (a a > ) C(k) π a e k /4a e ax e ikx dx π e a(x+ik/a) e k /4a dx e k /4a π π a a a e ax a e k /4a f(x, y, z) f(x, y, z) ( π ) 3 ( π ) 3 C(k x, k y, k z )e ixkx e iyky e izkz dk x dk y dk z C( k)e i k r d 3 k, k (kx, k y, k z ), r (x, y, z)

166 f(x) { ( π < x < ) ( x < π). f(x) sin x 3. f(x) x ( π < x < π) (a) f(x) (b) x π : k 4. f(x) x ( π x π) (a) f(x) (b) : k k π 6 (k + ) π 8 5. a 6. f(x), g(x) f(x) π f(x) { a ( x a ) ( x > a ) (a > ) F (k)e ikx dk, g(x) π h(x) h(x) π f(x t)g(t)dt G(ω)e iωx dω 7. f(x) F (k) f(x) dx F (k) dk

167 C n f(x) n π C n π π [ e inx π in C n e inx + n f(x)e inx dx π e inx dx π n ] π (n ) i πn [( )n ] (n ) i πn [( )n ]e inx + + i πn [( )n ]i sin(nx) + + π m sin[(m + )x] (m + ) m n i πn [( )n ]e inx ( i) i sin[(m + )x] π(m + ). C n π sin x e inx dx π π π sin x {cos(nx) i sin(nx)}dx π π ( sin x cos x sin x sin x ) C C π (n ±) C n π π π sin x cos(nx)dx sin x cos(nx)dx π π ( ) π {sin[( + n)x] + sin[( n)x]}dx π π π sin(x)dx, {[ ] π [ cos( + n)x cos( n)x + ( + n) ( n) { ( ) n+ + } ( ) n + n n ] π } [ + ( )n ] π ( n ) n C n sin x k π + π[( (k) ] eikx k k + π[( (k) ] eikx + π k π( 4k ) [e ikx + e ikx ] π + 4 cos(kx) π( 4k ) k

168 (a) C n π x e inx dx n C π π π π π π π x dx π3 n 3 C n x e inx dx x cos(nx)dx π π π ([ x ] π sin(nx) ) π x sin(nx) π n n ([ ] π x cos(nx) + ) π cos(nx)dx nπ n n n ( )n x π 3 + n (b) x π n ( )n [e inx + e inx ] π 3 + π π 3 + k k k 4 n 4 k ( )k cos(kπ) π 3 + k [ ] π π π n ( )n cos(nx) 4 k 4. (a) C π x dx π x dx π xdx π n π π π π C n π π π π x e inx dx π x [cos(nx) i sin(nx)]dx π π x cos(nx)dx π [ x sin(nx) ] π π n π πn [( )n ] π sin(nx) dx + n πn [ cos(nx) n ] π n C n x π + k π (b) (a) x k k π(k + ) [e i(k+)x + e i(k+)x ] 4 cos[(k + )x] π(k + ) π π (k + ) 8 k 4 π(k + )

169 f(x) π C(k) C(k)e ikx dk π f(x)e ikx dx π /a a /a cos(kx)dx π a k k lim a /a ae ikx dx a π [ k sin(kx) ] /a ( ) a k π k sin sin(k/a) lim a a π (k/a) π /a /a adx π k π ( ) a k π k sin a π e ikx dk e ikx dk δ(x) π π 6. π h(x)e ikx dx f(x t)g(t)e ikx dtdx π ( ) f(x t)e ik(x t) g(t)e ikt dxdt π (x t x ) ( π f(x )e ikx dx ) ( π g(t)e ikt dt ) F (k)g(k) (convolution) 7. f(x) π F (k)e ikx dk π f(x) dx ( π ) ( ) F (k)e ikx dk F (w) e iwx dw dx π F (k)f (w) e i(k w)x dxdkdw F (k)f (w) δ(k w)dkdw F (k) dk F (k)f (k) dk

170 r (x, y, z) 3 x i x + y + z i ϕ( r) ϕ( r) ρ( r) [ ] + µ ϕ( r) ρ( r) [ ] v t ψ( r, t) ρ( r, t) [ ] ψ( r, t) ρ( r, t) k t (v ) sin(ωt) x v x x/v sin[ω(t x v )] x x/v f(t x/v) g(t + x/v) f g t f(t x v ) f (t x v ), t f(t x v ) f (t x v ), x f(t x v ) v f (t x v ), x f(t x v ) v f (t x v ) [ v t f x f ] v t x f g v ϕ(x, t) [ ] v t x ϕ(x, t)

171 5 65 η ξ ϕ(x, t) Aη(x vt) + Bξ(x + vt) A, B 3 [ ] [ v t x y ] z ϕ(x, y, z, t) v t x + y + z ϕ (ϕ/t ) ϕ sin[ω(t x/v)] Ψ(x, t) e iω(t x v ) c ( 3 m/s) ν ω πν ω λ c, ν c νλ ( ) E hν hω (h h h/(π)) E cp Ψ(x, t) Ψ(x, t) e iω(t x c ) e ī h (Et px) E mv p /(m) i t e h (Et px) i Ē h e ī h (Et px), x e i h (Et px) p h e i h (Et px) i h h Ψ(x, t) t m Ψ(x, t) x 3 ] h [i h + t m Ψ(x, y, z, t) E c p + m c 4 p E mc Ψ(x, t) E Ψ(x, t) t h

172 5 66 h [ t Ψ(x, t) h c x Ψ(x, t) + m c 4 Ψ(x, t) h c x m c ] h Ψ(x, t) 3 [ c t + m c ] h Ψ(x, y, z, t) Ψ (Ψ/t ) 5. [ ] v t ψ( r, t) ρ( r, t) [ ] v t G( r, t ; r, t ) δ 4 ( r r, t t ) δ(x x )δ(y y )δ(z z )δ(t t ) G( r, t ; r, t ) ρ( r, t) ψ( r, t) ρ( r, t )G( r, t ; r, t )d 3 r dt ψ( r, t) [ ] v t ψ( r, t) [ ] v t ρ( r, t )G( r, t ; r, t )d 3 r dt [ ρ( r, t ) ] v t G( r, t ; r, t )d 3 r dt ρ( r, t )δ 4 ( r r, t t )d 3 r dt ρ( r, t) G( r, t ; r, t ) 5.3 c h ( ) [ m ]G( r) δ 3 ( r) : G( r) (π) 3/ G( k)e i k r d 3 k δ 3 ( r) (π) 3 G( k)[ k + m ]e i k r d 3 k (π) 3/ (π) 3 e i k r d 3 k e i k r d 3 k

173 5 67 G( k) G( r) (π) 3/ (π) 3/ [ k G( r) + m ] (π) 3/ [ k e i k r d 3 k + m ] (π) 3 π π (k + m ) eikr cos θ k sin θdϕdθdk (k k, r r, θ k r ) ϕ θ cos θ s sin θdθ ds G( r) (π) 4π ri k (k + m ) eikrs dsdk 4π k (k + m ) [eikr e ikr ]dk 4π ri k (k + m ) [ ] e ikrs s ikr s dk k i sin(kr)dk (k + m ) I ik (k + m ) sin(kr)dk C + C C+C ze irz z + m dz C z m x 8 C 8 C+C ze irz z + m dz xe irx π dx + lim x + m R ix sin(rx) x + m dx + z ±im C+C ze irz dz πires z + m [ ze irz z + m Re iξ R e iξ + m eirr(cos ξ+i sin ξ) ire iξ dξ ix sin(rx) x + m dx I ] zim G( r) 4π ri iπe mr 4πr e mr iπe mr m ( ) m ( ) 4πr

174 [. x ] v t f(x, t) f(x, t) f(k, ω) k ω f(x, t) ( π) f(k, ω)e ikx e iωt dkdω. G(x, t) : [ x ] G(x, t) δ(x)δ(t). µ t µ G(x, t) G(k, ω) G(x, t) δ(x)δ(t) ( G(k, ω)e ikx e iωt dkdω, π) ( ) ( ) e ikx dk e iωt dω. π π 3. G(x, t) 4π e ikx e iωt [k ( iω µ ) ] dkdω (µ > ) : (a) (b) ω. G(x, t) µ π θ(t)e µ k t e ikx dk. θ(t) t t < k. G(x, t) θ(t)µ tπ e x /(4µ t) 4. 3 Green G : [ x + y + x ] c t G(x, y, z, t) δ(x)δ(y)δ(z)δ(t) c (a) G(x, y, z, t) G(k x, k y, k z, ω) G(x, y, z, t) ( π) 4 G(kx, k y, k z, ω)e ikxx e ikyy e ikzz e iωt dk x dk y dk z dω (b) G(k x, k y, k z, ω) G(x, y, z, t)

175 f(x, t) ( π) [ x ] [ v t f(x, t) x ] v t ( π) [ ( f(k, ω) π) x ] v t e ikx e iωt dkdω ( f(k, ω)[(ik) π) v (iω) ]e ikx e iωt dkdω ( f(k, ω)[k ω π) v ]eikx e iωt dkdω f f(k, ω)e ikx e iωt dkdω f(k, ω)e ikx e iωt dkdω. : k ω v ω ±vk [ x ] G(x, t) δ(x)δ(t). µ t [ x ] ( ) ( ) µ t ( G(k, ω)e ikx e iωt dkdω e ikx dk e iωt dω π) π π [ G(k, ω) π x ] e ikx e iωt dkdω e ikx e iωt dkdω µ t 4π [ G(k, ω) (ik) ( iω) ] e ikx e iωt dkdω e ikx e iωt dkdω π µ 4π [ G(k, ω) k (iω) ] e ikx e iωt dkdω e ikx e iωt dkdω π µ 4π 3. (a) G(x, t) 4π iµ 4π G(k, ω) ( ) π k (iω) µ e ikx e iωt iµ dkdω [k (iω) ] 4π µ [ e iωt ] [ω + iµ k ] dω e ikx dk e ikx e iωt [ω + iµ k ] dkdω [ ] ω y e iωt [ω + iµ k ] dω e ity [ y + iµ k ] ( dy) e ity [y iµ k ] dy

176 (b) 8 x y, a t, ϵ µ k e ity [y iµ k ] dy θ(t)πie µ k t G(x, t) iµ 4π G(x, t) µ π [ θ(t)πie µ k t ]e ikx dk µ θ(t)e µ k t e ikx dk π µ π θ(t) θ(t)e µ k t e ikx dk µ π θ(t) µ π θ(t)e x /(4µ t) µ π θ(t)e x /(4µ t) e µ t{k ix/(µ t)} x /(4µ t) dk π e µ t{k ix/(µ t)} dk µ t θ(t)µ tπ e x /(4µ t) e µ k t e ikx dk ( k 4 ) (a) G [ ] c t G(x, y, z, t) [ ] G(kx c t, k (π) y, k z, ω)e ikxx e ikyy e ikzz e iωt dk x dk y dk z dω G(kx, k (π) y, k z, ω) [ ] c t e ikxx e ikyy e ikzz e iωt dk x dk y dk z dω [ ] G(kx, k (π) y, k z, ω) k ( ω ) e ikxx e ikyy e ikzz e iωt dk c x dk y dk z dω δ(x)δ(y)δ(z)δ(t) (π) 4 G(k x, k y, k z, ω) (π) k (ω /c ) e ikxx e ikyy e ikzz e iωt dk x dk y dk z dω

177 5 7 (b) G(x, y, z, t) ( π) 4 (π) k (ω /c ) eikxx e ikyy e ikzz e iωt dk x dk y dk z dω k x, k y, k z m ω /c m ±iω/c ( ) /(π) x + y + z r (π) 4 ω k (ω /c ) eik xx e ikyy e ikzz e iωt dk x dk y dk z (π) G(x, y, z, t) (π) 4πr e±iωr/c e iωt dω (4πr) (π) 4πr δ(t ± r c ) 4πr e±iωr/c e iωt e iω(t±r/c) dω

178 IV B ( )

179 f(x) (f (x), f (x), f (x)...,) F (x, f, f, f,...) f f n n : d f(x) kf(x) (k ) dx : m d dt x(t) mω x(t) η d x(t) (m, ω, η ) dt n n (f(x ) f, f (x ) f, f (x ) f,...) d f(t) G(f(t), t) dt t t f(t ) f (f ) t t f(t + t) f(t ) + f (t ) t + O( t ) f + G(f, t ) t + O( t ) f, t, t f(t + t) f(t + t), f(t + 3 t),... t f(t) d f(t) dt P (f(t), g(t), t) d g(t) dt Q(f(t), g(t), t) t t f(t ) f, g(t ) g f(t + t) f(t ) + P (f(t ), g(t ), t ) t + O( t ) f + P (f, g, t ) t + O( t ) g(t + t) g(t ) + Q(f(t ), g(t ), t ) t + O( t ) g + Q(f, g, t ) t + O( t )

180 6 74 f(t + t), g(t + t) t f(t), g(t) 3 d dt f (t) G (f (t), f (t),..., f n (t), t) d dt f (t) G (f (t), f (t),..., f n (t), t). d dt f n(t) G n (f (t), f (t),..., f n (t), t) t t f (t ) f, f (t ) f,..., f n (t ) f n, t f (t), f (t),..., f n (t) d df(t) f(t) G(f(t), dt dt, t) df(t) dt g(t) d d f(t) g(t), dt g(t) G(f(t), g(t), t) dt t f(t) n n n 6. dy dx y x + C (C ) 3 y C x y C 3 dy dx y x 3 3 x

181 6 75 dy dx y x y dy x dx 3 y ln y ln x + C ln ec x (C ) y A x (A e C ) 3 3 x x y y x md /dt r(t) F ( r, d r/dt, t) r(t) x, y, z 3 6 r(t) 6 t r(t ) v(t ) d r/dt tt ( ) m v θ ( < θ < π/) g x z t r(t) (x(t), y(t), z(t)) z m d d x(t) m dt dt y(t) m d z(t) mg dt O v θ x x(t) A t + A, y(t) B t + B (A,, B, ), z(t) gt + C t + C (C, C ) f df/dt x() y() z() ẋ() v cos θ, ẏ(), ż() v sin θ A B B C A v cos θ C v sin θ x(t) (v cos θ)t, y(t), z(t) gt + (v sin θ)t

182 d d x(t) ω y(t), dt y(t) ω x(t) (ω ) dt x(t) y(t). m d dt x(t) kx(t) + qe µ d dt x(t) (m, q, E, µ ) p(t) x(t) 3. d dr V (r) q (q ) V ( ) r 4. m d dt x(t) mg k d x(t) (m, g, k ) dt x() h ẋ() (hint: (dx(t)/dt) + (mg/k) f(t) ) 5. (a) z f(x) (b) (c) z x θ (d) t (L,, ) (L > ) v θ

183 d dt x(t) ω y(t) t d y(t) ω x(t) dt d dt x(t) ω d dt y(t) ω(ω x(t)) ω x(t) d dt y(t) ω y(t) t d x(t) ω y(t) dt d dt y(t) ω d dt x(t) ω( ω y(t)) ω y(t) d dt x(t) ω x(t), d dt y(t) ω y(t). p(t) p(t) m d x(t) dt d d p(t) m dt dt x(t) kx(t) + qe µ d dt x(t) kx(t) + qe µ m p(t) 3. r d dt p(t) µ p(t) kx(t) + qe m d dt x(t) m p(t) d dr V (r)dr q r dr V (r) q r + C (C ) V ( ) lim r ( q r + C) C V (r) q r 4. (dx(t)/dt) + (mg/k) f(t) m d f(t) dt d m dt x(t) mg k d x(t) kf(t) dt d f(t) dt f(t) k m d dt ln f(t) k m

184 6 78 t d dt ln f(t)dt k m dt ln f(t) k t + C (C ) m f(t) e C kt/m e C e kt/m De kt/m (D e C ) f(t) t d mg x(t) + De kt/m dt k ( ) d mg x(t) + dt De kt/m dt dt k x(t) + mg k t m k De kt/m + E (E ) x(t) mg k t + C e kt/m + C (C, C ) x() h ẋ() C, C h x() C + C ẋ() mg k k m C C m g, C k h C h + m g k x(t) mg k t m g k e kt/m + h + m g k h mg k t + m g k ( e kt/m ) ( ) 5. (a) x(t) (v cos θ)t t x z(t) v cos θ z gt + (v sin θ)t g( x x v cos θ ) + (v sin θ) v cos θ g v cos θ x + (tan θ)x (b) g z v cos θ x + (tan θ)x ( g x v cos ( θ g v tan θ) + cos ) θ tan θ v cos θ g v cos θ g ( ) g x v sin θ + v sin θ v cos θ g g

185 6 79 z (c) (a) z x v sin θ g ( ) g g v cos θ x + (tan θ)x x tan θ v cos θ x (d) L x tan θv cos θ g v sin θ cos θ g v sin θ g L v sin θ g v sin θ gl v θ

186 d f(x) P (x)q(f) dx P, Q Q x df Q(f) dx dx P (x)dx Q(f) df P (x)dx f f x x /Q P f(x) x dy ( ) dx x y ydy xdx y x + C (C ) x + y C ( C ) 7. d dx y(x) P (y x ) P y/x f f + x df dx dy dx d df (xf) f + x dx dx df P (f) dx P (f) f x

187 7 8 P (f) f df x dx f(x) y xf(x) y(x) ( ) x y + xy dy dy dx dx y x (y/x) xy (y/x) y xf f + x df dx ( f ) f x df dx (f + ) f f f + df x dx ln(f + ) ln x + C (C ) ln( y + ) + ln x C x x( y x + ) C ( C ) x + y Cx 7.3 d f(x) + P (x)f(x) Q(x) dx P, Q f, f Q f f d dx f d (x) + P (x)f (x), dx f (x) + P (x)f (x) c, c c f + c f [ ] [ ] d d d dx [c f + c f ] + P (x)[c f + c f ] c dx f + P (x)f + c dx f + P (x)f 7.3. d f(x) + P (x)f(x) dx f df P (x)dx ln f P (x)dx + C f(x) Ce P (x)dx ( C ) (C )

188 d dx f(x) + P (x)f(x) Q(x) e P (x)dx e P (x)dx d dx f(x) + e P (x)dx P (x)f(x) e P (x)dx Q(x) d [ e P (x)dx f(x) ] e P (x)dx Q(x) dx [e e P (x)dx f(x) P (x)dx Q(x) ] dx + C (C ) { [e f(x) e P (x)dx P (x)dx Q(x) ] } dx + C g(x) f(x) d d g(x) + P (x)g(x) Q(x), f(x) + P (x)f(x) dx dx f(x) + g(x) d [f + g] + P (x)[f + g] + Q(x) Q(x) dx f(x) + g(x) Ce P (x)dx + g(x) C d ( ) dt v(t) + kv(t) ae t (k, a ) v(t) be t (b ) d dt [be t ] + kbe t ae t ( b + kb)e t ae t a k b (k ) k a (k ) e t d v(t) + kv(t) dt Ce kt (C ) a v(t) (k ) e t + Ce kt k v(t) f(t)e t d dt [f(t)e t ] + f(t)e t ae t df dt e t f(t)e t + f(t)e t ae t df dt a f at + C (C ) v(t) (at + C)e t

189 N() N ( ) d dt N(t) τ N(t) (τ ). m v m c v m v c m v m dv dm dv dm c m mv m(c v) + (m m)(v + v) mc + m v m v ( m dm m ) m ϵm ( < ϵ < ) 3. r() cos θ + r sin θ dθ dr 4. xy (x y ) dy dx xy 5. E(t) E e kt (V) (k > ) R (Ω) C (F) Q I I dq dt E(t) RI + Q C R d Q(t) Q(t) + dt C krc Q() E(t) R C

190 d dt N(t) τ N(t) dn(t) dt N(t) dt τ dt N dn τ dt ln N t τ + C (C ) N(t) e C e t/τ N() N N N() e C N(t) N e t/τ. dv dm c m m dv dm dm c m dm v(m) c ln m + A (A ) 3. v(m ) A c ln m v(m) c ln m + c ln m c ln( m m ) ( ϵ)m r m v(( ϵ)m ) c ln( ) c ln( ϵ) ( ϵ)m cos θ + r sin θ dθ dr + r tan θ dθ dr tan θ dθ dr r tan θ dθ dr dr r dr tan θdθ ln r + C (C ) ln(cos θ) ln r + C r A cos θ (A ) r() r() A cos A r(θ) cos θ θ(r) arccos(r)

191 dy dx xy x y (y/x) (y/x) y/x f dy dx f + x df dx f y/x f + x df dx f f x df dx f f f f + f 3 f ( f ) (f 3 + f) df x dx [ f f ] df ln x + C (C ) f + ln f ln(f + ) ln x + C ( ) f ln ln x + C f + f Ax (A ) f + (y/x) (y/x) + Ax xy x + y Ax x + y qy (q /(A) ) x + (y q) q (, q) q 5. d Q(t) Q(t) + dt RC E R e kt Q(t) Xe kt (X ) kxe kt + X RC e kt E R e kt ( ) RC k X E R X RC ( krc) E R CE ( krc) CE Q (t) ( krc) e kt d Q(t) Q(t) + dt RC Q (t) Q (t) Ae t/(rc) (A )

192 7 86 Q (t) Q (t) Q(t) Ae t/(rc) + Q() CE ( krc) e kt (A ) Q() A + CE ( krc) A CE ( krc) Q(t) CE ( krc) (e kt e t/(rc) )

193 87 8 t r(t) m F F t r(t) v(t) ( ) m d dt r(t) F (t, r, v, K) (K ) x 8. (mg) (qe) F f (f ) ( f ) m d dt x(t) f. x(t) f m t + C t + C C C t x(t ) ẋ(t ) (ẋ dx/dt) 8. µ dx dt (µ ) dx dt v(t) m m d x(t) f µdx dt dt d v(t) f µv(t) dt v(t) f µ h(t) d dt h(t) µ m h(t)

194 8 88 h(t) C e µt/m v(t) C e µt/m + f µ (C ) t x(t) mc µ e µt/m + f µ t + C C e µt/m + f µ t + C ( C C ) 8.3 x k (k > ) m d x(t) kx(t) d dt dt x(t) ω x(t) k ω m ω sin(ωt) cos(ωt) C C x(t) C cos(ωt) + C sin(ωt) C + C sin(ωt + ϕ) ( sin ϕ C C + C, cos ϕ C C + C ) π/ω x(t) Ce λt (C, λ ) λ Ce λt ω Ce λt λ ω λ ±iω x(t) C e iωt + C e iωt ( C C ) x(t) x(t) C C 8.4 ( ) m d dt x(t) kx(t) µ d d x(t) m dt dt x(t) + µ d x(t) + kx(t) dt ẍ ẋ x ( ) x (t) x (t) C x (t) + C x (t) (C C )

195 8 89 x(t) e λt (mλ + µλ + k)e λt mλ + µλ + k λ i) µ 4mk < m [ µ ± i 4mk µ ] µ 4mk µ m α ω λ m α ± iω x(t) C e αt+iωt + C e αt iωt e αt [ C e iωt + C e iωt ] (C C ) [ ] e αt ii) µ 4mk > µ 4mk β λ α ± β m [ ] x(t) C e αt+βt + C e αt βt e αt C e βt + C e βt (C C ) iii) µ 4mk λ µ m teλt m d dt (teλt ) + µ d dt (teλt ) + k(te λt ) m(λe λt + λ te λt ) + µ(e λt + λte λt ) + k(te λt ) (mλ + µλ + k)(te λt ) + (mλ + µ)e λt x(t) (C t + C )e µt/(m) (C C ) 8.5 ( ) f cos(γt) (f γ ) m d x(t) kx(t) + f cos(γt) dt m d x(t) kx(t) dt x(t) A cos(γt) maγ cos(γt) ka cos(γt) + f cos(γt) (k mγ )A f

196 8 9 ω k m γ A f m(ω γ ) x(t) C cos(ωt) + C sin(ωt) + ω γ d dt z(t) ω z(t) + f m eiωt f m(ω γ ) cos(γt) z(t) y(t)e iωt [ÿ + iωẏ ω y]e iωt ω ye iωt + f m eiωt ÿ + iωẏ f m y Ct y if mω t Re [ if ] mω teiωt f mω t sin(ωt) x(t) C cos(ωt) + C sin(ωt) + f mω t sin(ωt)

197 ( g) (a) (b) v (a) (c) (d) (e) h (d). x(t) C e iωt + C e iωt ( C C ) x() x ẋ() v C C 3. x() A > ẋ() (a) µ 4mk < (b) µ 4mk > (c) µ 4mk 4. µ 4mk < x(t) 5. x() ẋ() v > ω γ ϵ ω + γ 6. R (Ω) L (H) C (F) I Q I dq dt RI + L di dt + Q C E R t R di dt + I Ld dt + I C de dt (a) E V ( ) I() Q() I (b) E(t) V cos(ωt) I I cos(ωt + ϕ) I ϕ E(t) C L

198 (a) m mg k (k > ) z t z(t) m d z(t) dt mg k (b) dz dt v ( dz(t) m dv dt mg kv dv dt k ( v + mg ) m k dt ) v + (mg/k) dv k m dt v mg mg k tan θ dv k cos θ dθ mg (mg/k)(tan θ + ) k cos θ dθ k t + C (C ) m k dθ k mg m t + C kg θ m t + C (C ) v(t) mg kg k tan m t + C z(t) mg kg v(t)dt tan k m t + C dt m kg k ln cos m t + C + C (C ) C, C v v() ( ) mg k k tan C C arctan v mg z() m k ln cos C + C C m k ln cos C

199 8 93 (c) v t mg kg k tan m t + C t m kg C z z(t ) m kg k ln cos m t + C + C C (d) (a) m d z(t) dt mg + k ( dz(t) (e) (b) k k k v (mg/k) dv m dt mg mg v k tanh u dv k cosh u du mg (mg/k)(tanh u ) k cosh u du k t + C (C ) m k du k mg m t + C kg u m t + C (C ) v(t) mg kg k tanh m t + C z(t) mg kg v(t)dt tanh k m t + C dt m kg k ln cosh m t + C + C (C ) C, C v() dt ) mg k tanh C C h z() m k ln cosh C + C C h. x(t) C e iωt + C e iωt x() x ẋ() v x x() C + C v ẋ() iω (C C ) C, C C ( x i v ) ω, C ( x + i v ) ω

200 x(t) e αt [ C e iωt + C e iωt ] (µ 4mk < ) (C t + C )e µt/(m) (µ 4mk ) [ ] e αt C e βt + C e βt (µ 4mk > ) α µ 4mk µ m, ω (µ µ 4mk 4mk < ), β m m x() A > ẋ() (µ 4mk > ) (a) µ 4mk < A C + C, (iω α)c (iω + α)c C (iω + α)a (iω α)a, C iω iω [ (iω + α)a x(t) e αt e iωt + iω (b) µ 4mk A C, C µ m C C µ m A, C A ] [ (iω α)a e iωt Ae αt cos(ωt) + α ] iω ω sin(ωt) x(t) A( µ m t + )e µt/(m) (c) µ 4mk > A C + C, (β α)c (β + α)c (α + β)a ( α + β)a C, C β β [ (α + β)a x(t) e αt e βt + β ] [ ( α + β)a e βt Ae αt cosh(βt) + α ] β β sinh(βt) (b) (a) (c) 4. e αt [ C e iωt + C e iωt ] [ e πα/ω ] π ω

201 x(t) C cos(ωt) + C sin(ωt) + x() ẋ() v > C + f cos(γt) (ω γ) m(ω γ ) f m(ω γ ), v ωc x(t) f m(ω γ ) cos(ωt) + v ω sin(ωt) + f m(ω γ ) cos(γt) f mϵ(ω ϵ) cos(ωt) + v ω sin(ωt) + f cos((ω ϵ)t) mϵ(ω ϵ) f( + (ϵ/ω)) cos(ωt) + v f( + (ϵ/ω)) sin(ωt) + (cos(ωt) + ϵt sin(ωt)) 4mϵω ω 4mϵω v ω sin(ωt) + ft [ mω sin(ωt) v ω + ft ] sin(ωt) mω 6. (a) E V ( ) de dt L d I dt + RdI dt + I C m L, µ R, k I() Q() C RI + L di dt + Q C V di V dt t L (b) E V cos(ωt) de dt V ω sin(ωt) L d I dt + RdI dt + I C V sin(ωt) I I cos(ωt + ϕ)

202 8 96 ω LI cos(ωt + ϕ) ωri sin(ωt + ϕ) + I C cos(ωt + ϕ) V ω sin(ωt) ( L) C ω cos(ωt + ϕ) ωr sin(ωt + ϕ) V ω sin(ωt) I ( ) { ωc ωl + R } sin(ωt + ϕ + θ) V sin(ωt) I sin θ ( ωc ωl) ( ωc ωl) + R t, cos θ R ( ωc ωl) + R I V ( ωc ωl) + R, ϕ θ

203 q v B q v B B (,, B) m d dt r(t) q d dt r B m d x dy dt qb dt, y dx md qb dt dt, z md dt z z(t) C z t + C z, (C z C z ) x y qb m dv x dt qb m v y, ω t dv y dt qb m v x d v x dt ω dv y dt d v y ω v x, dt ω dv x dt ω v y v() (v x, v y, v z ) v x (t) v x cos(ωt) + C x sin(ωt), v y (t) v y cos(ωt) + C y sin(ωt), v z (t) v z C x v y C y v x r() (,, ) x(t) v x ω sin(ωt) + v y ω [cos(ωt) ], y(t) v y ω sin(ωt) v x ω [cos(ωt) ], z(t) v zt v x + vy xy ω (v y, v x ) v z ω z

204 k m d x dt kx H H mv kx v p mv H p x H p m + kx p m dx dt, dp dt kx dx dt H p, dp dt H x H( ) ( ) ẋ p, ṗ kx xp m xp H p m + kx ( ) xp ( ) p O x (x, y, z) (p x, p y, p z ) 6 n 6n 6n

205 z ( ) x r cos θ y r sin θ z z r e r θ e θ x y e x e y y r e θ e r ( er e θ ) ( cos θ sin θ sin θ cos θ ) ( ex e y ) O θ x r θ e r e θ { x r cos θ y r sin θ { ẋ ṙ cos θ r sin θ θ ẏ ṙ sin θ + r cos θ θ v d r dt ẋ e x + ẏ e y (ṙ cos θ r sin θ θ) e x + (dotr sin θ + r cos θ θ) e y ṙ(cos θ e x + sin θ e y ) + r θ( sin θ e x + cos θ e y ) ṙ e r + r θ e θ { ẍ r cos θ ṙ sin θ θ r cos θ( θ) r sin θ θ ÿ r sin θ + ṙ cos θ θ r sin θ( θ) + r cos θ θ a d r dt ẍ e x + ÿ e y [ r cos θ ṙ sin θ θ r cos θ( θ) r sin θ θ] e x + [ r sin θ + ṙ cos θ θ r sin θ( θ) + r cos θ θ] e y ( r r θ ) e r + (ṙ θ + r θ) e θ ( r r θ ) e r + d r dt (r θ) eθ F e r e θ F F x e x + F y e y F x (cos θ e r sin θ e θ ) + F y (sin θ e r + cos θ e θ ) (F x cos θ + F y sin θ) e r + ( F x sin θ + F y cos θ) e θ F r e r + F θ e θ m( r r θ ) F r, m r d dt (r θ) Fθ F θ mr θ l l z θ l mr ( ) l m[ r r ] m[ r l mr m r ] F 3 r

206 m q > v ( ) B d L q v B O d L. ( F mω r) ( m q) B (,, B) (a) (b) z ( ) (c) x A cos(ωt + δ) y A sin(ωt + δ) ω ω 3. x p xp 4. (µ 4mk < ) x() A > ẋ() x p xp 5. xy r A (A ) r ( ) 6. F θ mr θ l u r (a) ṙ du dθ l (b) r u d u dθ l (c) r m u d u dθ l (d) E U F r d U(r) dr E ( ) du + u m (E U) dθ l

207 z B (,, B) t r() (,, ), r() (v,, ) x(t) v ω sin(ωt) y(t) v [cos(ωt) ] ω z(t) x d t d x(t ) v ω sin(ωt ), ω qb m d v/ω mv/(qb) x > d d < mv/(qb) sin(ωt ) dω v y(t ) v ω [cos(ωt ) ] v ω [ ( dω v ) ] z(t ) t v x (t ) v cos(ωt ) v ( dω v ) v y (t ) v sin(ωt ) dω v z (t ) x d + L t + t v x (t )t L t L v x (t ) L v ( dω v ) t + t y y(t ) + v y (t )t v ω [ ( dω L v ) ] + dω v ( dω v ) ( dw v ) v ω dωl v (dω v ) + dωl v [ + (dω v ) ] + [ v ω + dωl v ](dω v )

208 9. (a) m d r dt mω r + q d r dt B m d x dt mωx + q( dy dt B z dz dt B y) mωx + qb dy dt m d y dt mωy + q( dz dt B x dx dt B z) mωy qb dx dt m d z dt mω z + q( dx dt B y dy dt B x) mω z (b) z z(t) C z e iω t + C z e iω t (C z, C z ) (c) x, y x A cos(ωt + δ) y A sin(ωt + δ) ω A cos(ωt + δ) ω A cos(ωt + δ) + qbω m A cos(ωt + δ) ω A sin(ωt + δ) ω A sin(ωt + δ) + qbω m A sin(ωt + δ) A, A ( ω ω qbω m qbω ω ω m ) ( A A ) ( ) ω ω qbω m qbω m ω ω (ω ω) ω ω ± qbω m ( ) qbω m ( Zeeman ) 3. ( ) m d x dt dp dt mg, mg p mdx dt p mdx dt p(t) mgt + p, x(t) gt + p m t + x

209 9 3 p, x t t x ( ) g p p + p ( ) p p + x mg m mg g ( p p p ) + g ( ) p + x mg mg mg m g (p p ) + x p O x 4. x(t) e αt [ C e iωt + C e iωt ] (C C ) α µ m, ω 4mk µ m x() A > ẋ() A x() C + C ẋ() (iω α)c (iω + α)c iω(c C ) α(c + C ) C ( A i αa ), C ( A + i αa ) ω ω x(t) Ae αt [ cos(ωt) + α ω sin(ωt) ] p(t) mẋ(t) ma (ω + α ) e αt sin(ωt) ω p x

210 m[ r l m r 3 ] A r, l mr θ r ṙ r l A mr 3 r (mr θ) A mr 3 r ( r 3 A 4π m π θ ) A 4π m T (T π/ θ ) 6. (a) ṙ du u dt du u dθ θ du u dθ l mr l m du dθ (b) (c) r r d dtṙ l d u m dθ θ l d u lu ( ) lu m dθ m d u m dθ ( ) F r m( r r θ lu d ) m ( ) u l m dθ r l u [ d ] u mr m dθ + u (d) (c) F r du dr du dθ θ l u [ d ] u m dθ + u l [ d ] u m dθ + u [ l d ] u du m dθ + u dθ l ( ) d du + u m dθ dθ ( du dθ du dr du du du dr du du du du du dθ du dθ u du du ) + u m (C U) (C ) l

211 9 5 E E m (ṙ + r θ ) + U m E U ( ) l du + u m dθ ( ) du + u dθ m (E U) l l m ( ) du + dθ u ( ) lu + U m E C ( ) E

212 6 3 f (n) (x) + P (x)f (n ) (x) + + P n (x)f (x) + P n (x)f(x) Q(x) f(x) k a k (x a) k a k 3. P k (x) (k,..., n) Q(x) x a ( x a ) x a n f(x) a k (x a) k k d ( ) : dt x(t) + ω x(t) t t x(t) a k t k k k(k )a k t k + ω a k t k k k k s (s + )(s + )a s+ t s + ω a k t k s k s k k[(k + )(k + )a k+ + ω a k ]t k ω a k+ (k + )(k + ) a k k x(t) n [ ω a n (n)(n ) a n ( ) n ωn (n)! a ω a n+ (n + )(n) a n ( ) n ω n (n + )! a ( ) n ωn ω n (n)! a t n + ( ) n (n + )! a t n+ ] a cos(ωt) + a ω sin(ωt) a a /ω C C

213 3 7 Q(x) P k (x) (k,..., n) Q(x) x a (x a)p (x) (x a) P (x)... (x a) n P (x) x a f(x) (x a) s n k a k (x a) k s ( ) 4x d y dx + 4xdy dx + (x )y 4x x y x s d y dx + ( dy x ) x dx + y 4x n k a k x k 4x (k + s)(k + s )a k x k+s + 4x (k + s)a k x k+s + (x ) a k x k+s k k k [ {4(k + s)(k + s ) + 4(k + s) }ak x k+s + a k x k+s+] k (4s )a x s + {4(s + ) }a x s+ + k[{4(k + s + ) }a k+ + a k ]x k+s+ x 4s s ± {4(s + ) }a a s 4(k + ) 4k + 4k 4k(k + ) y(x) x / n a n 4(n + )n a n ( ) n n (n + )! a, a n+ ( ) n n (n + )! a x n a x / ( ) n ( ) x n+ a sin( x n (n + )! x ) s 4(k ) 4k 4k 4k(k ) a n 4(n)(n ) a n y(x) x / n ( )n n (n)! a, a n+ ( ) n n (n)! a x n a x / ( ) n ( ) x n a cos( x n (n)! x ) y(x) C x cos( x ) + C x sin( x )

214 d dx f(x) + (λ x )f(x) (λ ) ( ) x (λ x ) λ x f(x) e y(x) [y + (y ) ]e y x e y y + (y ) x ( x ) y ax n (x n ) n(n )ax n + a n x n x n > x x n x n n a ±/ x a / f(x) e x / f(x) u(x)e x / [ u xu ( x )u ] e x / + (λ x )ue x / u xu + (λ )u u u(x) e z(x) [z + (z ) xz + (λ )]e z z bx n bn(n )x n + b n x n bnx n + (λ ) x b n x n bnx n n b f(x) e x e x / e x / x u(x) k a k x k k(k )a k x k x ka k x k + (λ ) a k x k k k k [(k + )(k + )a k+ + ( k + λ )a k ] x k k x a k+ (k λ + ) (k + )(k + ) a k n n λ + u(x) (x ) x λ n + (n )

215 3 9 n : λ n : λ 3 n : λ 5 n 3 : λ 7 u(x) a f(x) a e x / u(x) a x f(x) a xe x / u(x) a ( x ) f(x) a ( x )e x / u(x) a (x 3 x3 ) f(x) a (x 3 x3 )e x / a a u(x) H n (x) n H n (x) ( ) n ( ) k (x) k (n)! (k)!(n k)! k n H n+ (x) ( ) n ( ) k (x) k+ (n + )! (k + )!(n k)! k

216 ( x )y xy + 6y y ax + bx + c a b c y(). d dt N(t) kn(t) (k ) N(t) a k t k k N() N 3. ( ) x d dx J(x) + x d dx J(x) + (x n )J(x) (n ) (a) J(x) x s k a k x k s ±n (b) s n a k (c) J(x) 4. d dr f(r) + d r dr f(r) + (λ r 4 )f(r) ( ) (a) f(r) r r f(r) e y(r) y(r) (b) y(r) ar n a n f(r) r (c) y(r) f(r) u(r)e y(r) u(r) (d) u(r) k a k r k a k (e) u(r) r ( )

217 y ax + bx + c y ax + b y a ( x )y xy + 6y ( x )(a) x(ax + b) + 6(ax + bx + c) ( a 4a + 6a)x + ( b + 6b)x + (a + 6c) 4bx + (a + 3c) x b a+3c y 3cx +c y() c a 3 y 3 x. N(t) a l t l dn(t) l dt la l t l d N(t) kn(t) l dt la l t l k a l t l l l (l s ) (l s ) + )a s+ t s(s s k a l t l k a s t s l s [(s + )a s+ + ka s ]t s s t (s + )a s+ + ka s a s k s a s ( k) s(s ) a s ( k)s s! N(t) ( k) s a t s s s! N() N a N a ( kt) s N(t) N s! s N e kt (a) r r d dr f(r) 4 f(r) f(r) e y(r) f (r) y e y y [y + (y ) ]e y [y + (y ) ]e y 4 ey y + (y ) 4

218 3 (b) (a) y ar n n, y nar n y n(n )ar n n(n )ar n + n a r (n ) 4 r n (n ) ( ) n n y (a) y n a 4 a ± y ± r y r ey e r/ r n a y r (c) f(r) u(r)e r/ d dr [u(r)e r/ ] [ du dr u]e r/ d dr [u(r)e r/ ] [ d u dr du dr + 4 u]e r/ [ d u dr du dr + 4 u]e r/ + r [du dr u]e r/ + ( λ r 4 )ue r/ d u dr + ( r )du (λ ) + u dr r (d) (c) u(r) k a k r k k(k )a k r k + ( k r ) ka k r k (λ ) + a k r k k r k k(k )a k r k + ka k r k ka k r k + (λ )a k r k k k k k ) + k]a k r k[k(k k + (λ k)a k r k k k(k + )a k r k + [λ (k )r])a k r k k k {k(k + )a k + (λ k)a k }r k k r ( r ) a k k λ k(k + ) a k (e) u(r) k a k r k r (d) n n λ a n a n+ a n+

219 V C ( )

220 x + y + z [ + V (x, y, z)]ψ(x, y, z) V (x, y, z) x F (x) y G(x) z H(x) V (x, y, z) F (x) + G(y) + H(z) Ψ(x, y, z) Ψ(x, y, z) P (x)q(y)r(z) x y z ( ) P (x)q(y)r(z) [ + V (x, y, z)]ψ(x, y, z) [ ] x + y + + F (x) + G(y) + H(z) P (x)q(y)r(z) z [ d ] P (x) P (x) dx + F (x) + [ d ] [ Q(x) d ] R(x) Q(y) dy + G(y) + R(z) dz + H(z) x y x x, y, z C x, C y, C z d P (x) dx + F (x)p (x) C x P (x), d Q(y) dy + G(y)Q(y) C y Q(y), d R(z) dz + H(z)R(z) C z R(z), C x + C y + C z. 3. x, y, z x, y x r cos θ, y r sin θ, z z ( r, θ π)

221 3 5 (r, θ, z) x r x r + θ x θ cos θ r sin θ r θ r y y r + θ y θ sin θ r + cos θ r θ x ( cos θ x r sin θ ) r θ r ( cos θ x r r sin θ ) + θ ( r θ x θ cos θ cos θ sin θ cos θ sin θ + r r θ r rθ + sin θ sin θ cos θ sin θ cos θ + r r r rθ r y ( sin θ y r + cos θ ) r θ r ( sin θ y r r + cos θ ) + θ r θ y sin θ sin θ cos θ r r + cos θ r cos θ sin θ + r r cos θ r sin θ r ) θ θ + sin θ r θ ( sin θ θ r + cos θ r sin θ cos θ + θ r rθ cos θ sin θ rθ r ) θ θ + cos θ r θ x + y + z r + r r + r θ + z [ + a]ψ(x, y, z) (a ) Ψ Ψ R(r)F (θ)g(z) [ r + r R(r)F (θ)g(z) [ d R R(r) dr + r r + ] r θ + z + a R(r)F (θ)g(z) ] dr + d F dr r F dθ + d G G dz + a r, θ 3 z 4 C, C [ d R R(r) dr + ] dr + d F r dr r F dθ C, d G G dz C, C + C + a G(z) B e cz + B e cz, B, B c C

222 3 6 r [ d R R(r) dr + r ] dr + C r d F dr F dθ r θ m r d R dr + r dr dr + (C r m )R, d F dθ m F 3 F F D e imθ + D e imθ (D, D ) θ, F (θ) F (θ + π), m R C r s R(r) J(s) s d J ds + sdj s + (s m )J 3.3 x, y, z x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ ( r, θ π, ϕ π) (r, θ, ϕ) (3 ) x r x r + θ x θ + ϕ x ϕ sin θ cos ϕ cos θ cos ϕ + r r θ sin ϕ r sin θ ϕ r y y r + θ y θ + ϕ y ϕ sin θ sin ϕ cos θ sin ϕ + r r θ + cos ϕ r sin θ ϕ r z z r + θ z θ + ϕ z ϕ cos θ r sin θ r θ x, x, z f f r + f r r + f r θ + [ r r (rf) + r sin θ θ cos θ f r sin θ θ + f r sin θ ϕ sin θ f ] f + θ r sin θ ϕ (f ) [ + a]ψ(x, y, z) (a ) f f R(r)Θ(θ)Φ(ϕ) [ d R ΘΦ dr + ] [ dr d Θ + RΦ r dr r dθ + cos θ ] dθ + RΘ d Φ r sin θ dθ r sin θ dϕ + arθφ { [ d r sin R θ R dr + ] dr + [ d Θ r dr Θ r dθ + cos θ ] } dθ + a + d Φ r sin θ dθ Φ dϕ

223 3 7 d Φ dϕ + m Φ Φ(ϕ) C ϕ e imϕ + C ϕ e imϕ (m ) d Θ dθ r [ d R R dr Θ [ d Θ dθ + r ] dr dr + cos θ sin θ C R + C Θ + ar C R ] dθ dθ m sin θ CΘ + cos θ ( ) dθ sin θ dθ + λ m sin Θ (λ C Θ ) θ cos θ s Θ(θ) L(s) d dθ ds d dθ ds sin θ d ds, d dθ sin θ d ds cos θ d ds ( ( s ) d L ds sdl ds + λ m s ) L

224 [ ] h m x + y Ψ(x, y) EΨ(x, y) (m, h, E ) (a) Ψ(x, y) P (x)q(y) P (x), Q(y) d P dx k P, d Q dy l Q, k + l me h (k, l ) (b) P () P (a) P (x) (c) Q( b/) Q(b/) Q(y). (x r cos θ, y r sin θ) r x + y, tan θ y x, y x r x, r y, θ x, θ r, θ y 3. (x r cos θ, y r sin θ) r θ x, y, x, y 4. (x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ) r x + y x + y + z, tan θ, z tan ϕ y r x, y, z x x, r y, r z, θ x, θ y, θ z, ϕ x, ϕ y, r, θ, ϕ 5. (x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ) r, θ, ϕ x 6. (x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ) r, θ, ϕ y 7. (x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ) r, θ, ϕ z 8. 5, 6, 7 f (f ) f f r + f r r + f r θ + cos θ f r sin θ θ + f r sin θ ϕ

225 (a) Ψ(x, y) P (x)q(y) [ ] h m x + y P (x)q(y) EP (x)q(y) [ h d ] P m dx Q + P d Q dy EP (x)q(y) P d P dx + Q d Q dy me h x y x, y k, l d P dx k P, d Q dy l Q, k + l me h (b) d P dx k P P (x) C e ikx + C e ikx (C, C ) P () P (a) C + C, C e ika + C e ika C (e ika e ika ) ic sin(ka) (C C ) sin(ka) k nπ (n ) a P (x) C sin( nπx ) (C n ) a (c) (b) Q(y) D e ily + D e ilx (D, D ) Q( b/) Q(b/) D e ilb/ + D e ilb/, D e ilb/ + D e ilb/ ( e ilb/ e ilb/ e ilb/ e ilb/ ) ( D D ) ( ) (D D ) e ilb/ e ilb/ e ilb e ilb i sin(lb) e ilb/ e ilb/ l jπ b (j )

226 3 D e ijπ/ + D e ijπ/ D ( i) j + D i j [( ) j D + D ]i j j D D D D Q(y). r x + y x, y r x x r y y tan θ y/x x tan θ y/x y 3. D cos( (q )πy b ) D sin( qπy b ) x + y x + y x x + y x r (q ) cos θ, y x + y y r sin θ x tan θ x (y x ) θ cos θ x y x θ x cos θ r sin θ r cos θ sin θ r y tan θ y (y x ) θ cos θ y x θ cos θ y r cos θ cos θ r r θ x r x θ x + y r y cos θ x + sin θ y x + y θ y x x + y x + r sin θ x + r cos θ y y x + x y y x + y y 4. r x + y + z x, y, z r x x r y x r z z x + y + z x + y + z x + y + z x r sin θ cos ϕ x + y + z r y r sin θ sin ϕ x + y + z r z x + y + z r cos θ r cos θ sin θ cos ϕ sin θ sin ϕ

227 3 tan θ x + y /z x x tan θ x ( x + y ) z θ cos θ x x z x + y θ x r sin θ cos ϕ cos θ r cos θr sin θ tan θ x + y /z y y tan θ y ( x + y ) z θ y cos θ y z x + y θ cos r sin θ sin ϕ θ y r cos θr sin θ cos θ cos ϕ r cos θ sin ϕ r tan θ x + y /z z z tan θ z ( x + y ) z θ x + y cos θ z z θ cos θ r sin θ z r cos θ sin θ r tan ϕ y/x x tan ϕ y/x y tan ϕ y/x z x tan ϕ x (y x ) ϕ cos ϕ x y x ϕ x r sin θ sin ϕ cos ϕ r sin θ cos ϕ sin ϕ r sin θ y tan ϕ y (y x ) ϕ cos ϕ y x ϕ cos ϕ y r sin θ cos ϕ cos ϕ r sin θ z tan ϕ z (y x ) ϕ cos ϕ z ϕ z

228 x r x r + θ x θ + ϕ x ϕ sin θ cos ϕ r x r x r ( x ) + θ x θ ( x ) + ϕ x ϕ ( x ) ( sin θ cos ϕ sin θ cos ϕ cos θ cos ϕ r r θ ( cos θ cos ϕ + cos θ cos ϕ + sin θ cos ϕ r r θr cos θ cos ϕ + r θ sin ϕ ( sin θ sin ϕ + sin θ cos ϕ r sin θ r ϕr + cos θ cos ϕ r + cos θ cos ϕ r + cos θ cos ϕ r θ sin ϕ r sin θ sin θ cos ϕ r rθ + cos θ sin ϕ + r sin θ ϕ sin ϕ r sin θ ϕθ cos ϕ r sin θ cos θ sin ϕ r ϕ sin ϕ r sin θ θ θ θϕ ϕ ) ϕ sin ϕ r sin θ sin θ cos ϕ r + cos θ cos ϕ r θ + sin ϕ r sin θ ϕ + sin θ cos θ cos ϕ r θr cos θ sin ϕ cos ϕ ϕ cos ϕ sin r sin θ θϕ r rϕ + (sin ϕ + cos θ cos ϕ) r r ( cos θ sin ϕ + sin θ cos θ ) cos ϕ r sin θ r θ ( ) sin ϕ cos ϕ + + cos θ sin ϕ cos ϕ sin ϕ cos ϕ r r sin + θ r sin θ ϕ y y r y r y r + θ y θ + ϕ y ϕ sin θ sin ϕ r r ( y ) + θ y θ ( y ) + ϕ y ϕ ( y ) ( sin θ sin ϕ sin θ sin ϕ cos θ sin ϕ r r θ ( cos θ sin ϕ + cos θ sin ϕ + sin θ sin ϕ r r rθ cos θ sin ϕ + r θ + cos ϕ ( sin θ cos ϕ + sin θ sin ϕ r sin θ r rϕ + cos θ sin ϕ r + cos θ sin ϕ r + cos θ sin ϕ r θ + cos ϕ r sin θ sin θ sin ϕ r ) ϕ rθ cos ϕ r sin θ θ cos θ cos ϕ r sin θ ϕ + cos ϕ r sin θ ϕθ sin ϕ r sin θ + cos θ cos ϕ r θ ϕ + cos ϕ r sin θ ϕ θϕ sin θ sin ϕ r + cos θ sin ϕ r θ + cos ϕ r sin θ ϕ + sin θ cos θ sin ϕ r rθ cos θ sin ϕ cos ϕ ϕ cos ϕ + + sin r sin θ θϕ r rϕ + (cos ϕ + cos θ sin ϕ) r r ) ) ϕ sin ϕ r sin θ ϕ + cos ϕ r sin θ ) rϕ ) rϕ

229 ( ( sin θ cos θ sin ϕ r sin ϕ cos ϕ r + cos θ ) cos ϕ r sin θ θ cos θ sin ϕ cos ϕ r sin θ sin ϕ cos ϕ r sin θ ) ϕ 7. z z r z r z r + θ z θ + ϕ z ϕ cos θ r sin θ r r ( z ) + θ z θ ( z ) + ϕ z ϕ ( z ) ( cos θ cos θ r + sin θ r θ sin θ ) r rθ sin θ ( sin θ + cos θ r r rθ cos θ r cos θ r + sin θ r θ θ cos θ sin r θ θ sin θ r rθ + sin θ r ) θ θ cos θ + sin r r θ 8. 5, 6, 7 x + y + z (sin θ cos ϕ + sin θ sin ϕ + cos θ) ( r cos θ cos ϕ + + cos θ sin ) ϕ + sin θ r r r θ ( sin ) ϕ + r sin θ + cos ϕ r sin θ ϕ + r (sin ϕ + cos θ cos ϕ + cos ϕ + cos θ sin ϕ + sin θ) ( r cos θ sin ϕ + sin θ cos θ cos ϕ + cos θ cos ϕ sin θ cos θ sin ϕ r sin θ r r sin θ r ) sin θ cos θ + r θ r + r r + r θ + cos θ r sin θ θ + r sin θ ϕ

230 [ + a]ψ(x, y, z) x + y + z (r, θ, z) r x d J dx + xdj dx + (x n )J (n ) n n n x d J dx + dj n + ( x dx x )J J, J x J(x) J(x) x s a k x k (a ) k d J dx a k (k + s)(k + x )x k+s, k dj dx a k (k + s)x k+s k a k (k + s)(k + s )x k+s + a k (k + s)x k+s + a k (x n )x k+s k k k a k [(k + s)(k + s ) + (k + s) n ]x k+s + a k x k+s+ k k a k [(k + s) n ]x k+s + a k x k+s k k k +, 3,... k, 3,... k k, (s n )a x s + [(s + ) n ]a x s+ + k[{(k + s) n }a k + a k ]x k+s x x s, x s+, x s+,... (s n )a, [(s + ) n ]a, {(k + s) n }a k + a k (k, 3,...) a s ±n [(±n + ) n ]a (±n + )a n (±n + ) a

231 3 5 s +n 3 a k (k + s) n (k + n) n k(k + n) k (k m) (k m + ) (m,,,...) [k m ] [k m + ] a m (n + m)m a m 4 (n + m)m a m ( ) 4 (n + m)m(n + m )(m ) a (m ) a m+ J(x) m ( )l l (n + m)... (n + m l + )m... (m l + ) a (m l) ( )m m n! (n + m)!m! a (n + m + )(m + ) a m ( ) m (n + m + )(n + m )... (n + 3)(m + )(m )... 3 a ( ) m m n! (n + m)!m! a x n+m n! n a n! n a J n (x) m m ( ) m ( ) x n+m (n + m)!m! ( ) m ( ) x n+m (n + m)!m! s n < {(k+s) n }a k +a k k(k n)a k +a k k a k k k n a n +a n a n a n a n 4 a a a (n ) n N n (x) Y n (x) H () ν (x) J ν (x) + in ν (x) H () ν (x) J ν (x) in ν (x) 3. J n ( ) (n+m)! Γ(n+m+) ( ) m ( ) x n+m J n (x) Γ(n + m + )m! m

232 3 6 n n J n (x) J (x), J (x), J (x) J (x) J (x) J (x) n /Γ( ) n l (l ) J n (x) J l (x) ml ( ) l m ) l+m ( ) m ( x Γ( l + m + )m! ) l+m ( ) m Γ( l + m + )m! ( x k ( ) k k!γ(l + k + ) ( x k ) l+k ( ) l J l (x) ( m l k ) ( ) l+k ( ) x l+k Γ(k + )(l + k)! 3.. x n J n (x) x n J n (x) d dx [xn J n (x)] d dx [xn m m ( ) m ( ) x n+m] Γ(n + m + )m! m ( ) m Γ(n + m)m! x n+m n+m xn m nx n J n (x) + x n J n(x) x n J n (x) ( ) m Γ(n + m)m! ( x ( ) m (n + m)x n+m Γ(n + m + )m! n+m ) n+m x n J n (x) nj n (x) + xj n(x) xj n (x) d dx [x n J n (x)] d ( ) m ( ) x n+m] ( ) m mx m dx [x n m Γ(n + m + )m! m Γ(n + m + )m! n+m ( ) m x m Γ(n + m + )(m )! ( ) k+ x k+ n+m Γ(n + k + )k! n+k+ m x n k ( ) k Γ(n + + k + )k! ( x k ) n++k x n J n+(x) nx (n+) J n (x) + x n J n(x) x n J n+ (x) nj n (x) + xj n(x) xj n+ (x) J n(x) [J n (x) J n+ (x)], n x J n(x) [J n (x) + J n+ (x)] J n+ (x) n x J n(x) J n (x) J (x) J (x) n J n (x)

233 f(x) J n (ax) ax t J n (t) t ax t d f dt + tdf dt + (t n )f a x d f df a + ax dx adx + (a x n )f x d f dx + df dx + (a x n x )f d df [x dx dx ] + (a x n x )f g(x) J n (bx) (a b) q p x d g dx + dg dx + (b x n x )g d dg [x dx dx ] + (b x n x )g g d df [x dx [x dg dx ] (a x n x )fg + (b x n x )fg dx ] f d dx d dx [xf g xfg ] (b a )xfg p q d dx [xf g xfg ]dx [xf g xfg ] q p (b a ) [axj n(ax)j n (bx) bxj n (ax)j n(bx)] q p (b a ) J n(x) n x J n(x) J n+ (x) q p aj n(ax)j n (bx) bj n (ax)j n(bx) q p q p xfgdx xj n (ax)j n (bx)dx [ n x J n(ax) aj n+ (ax)]j n (bx) J n (ax)[ n x J n(bx) bj n+ (bx)] aj n (ax)j n+ (bx) bj n (bx)j n+ (ax) xj n (ax)j n (bx)dx (b a ) [axj n(ax)j n+ (bx) bxj n (bx)j n+ (ax)] q p p, q J n (ap) J n (bq) a b b a + ϵ ϵ x [ lim axjn (ax){j n+ (ax) + ϵxj ϵ aϵ n+(ax)} (ax + ϵx){j n (ax) + ϵxj n (ax) }J n+ (ax) + O(ϵ ) ] [J n(ax)j n+(ax) J n(ax)j n+ (ax)] x a J n(ax)j n+ (ax) x [J n(ax){j n (ax) (n + ) ax J n+(ax)} { n ax J n(ax) J n+ (ax)}j n+ (ax)] x a J n(ax)j n+ (ax) J n (pa) J n (qa) p, q [ ] q x q xj n (ax)j n (ax)dx p {J n+(ax)} ( J n (pa) J n (qa) ) p

234 e z (t t ) t e z (t t ) e z t e z t ( k t n k m n m (m + n)! m! (z )m+n ( z )m ) ( k! (zt )k m m! ( z ) t )m m J n (z) e z (t t ) n ( ) m ( ) z n+m Γ(n + m + )m! J n (z)t n (t ) J n (z) ( ) J n (x + y)t n e (x+y) (t t ) e x (t t ) e y (t t ) J k (x)t k J m (y)t m n k m J k (x)j n k (y)t n (m n k ) n k t n ( ) J n (x + y) k J k (x)j n k (y) t e iθ t t eiθ e iθ i sin θ e iz sin θ n J n (z)e inθ 3.4 r f d f dr + ( ) df r dr + n(n + ) f (n,,,...) r rf g d dr f d g dg dr r r dr g r 3 d dr f d dr ( dg r dr g ) d g r 3 r dr dg r 3 dr + 3 g 4 r 5

235 3 9 d f dr + ( df r dr + ) n(n + ) f r d g r dr dg r 3 dr + 3 g 4 + ( dg r 5 r r dr ) ( g + r 3 d g r dr + ( ) dg r 3 dr + n(n + ) + /4 g r r [ d g r dr + ( ) ] dg r dr + (n + /) g r ) n(n + ) g r r g n n + / j n (z) π z J n+/(z) ( ) J / (x) Γ(x)Γ(x + ) π Γ(x) x J / (x) m m ( ) m ( ) x m+/ Γ(m + + /)m! ( ) m (m+) ( ) x m+/ πγ(m + ) πx m ( ) m ( x Γ(m + + /)Γ(m + ) m ( ) m (m + )! xm+ πx sin x ) m+/ j (x) sin x x j n(x) x Γ(z) Re[z] > Γ(z) t z e t dt z Γ(z) (z )Γ(z ) Γ(z ) Γ(z) z ( ) x, y B(x, y) B(x, y) (t sin θ ) π/ t x ( t) y dt π/ (sin θ) (x ) (cos)θ) (y ) sin θ cos θdθ (sin θ) x (cos θ) y dθ

236 3 3 Γ(x)Γ(y) 4 4 e t t x dt e s s y ds (t p, s q ) ( ) ( ) e p p x dp e q q y dq e (p +q ) p x q y dpdq (p r cos θ, s r sin θ ) π/ ( e r r (x+y ) (cos θ) x (sin θ) y rdθdr e r r (x+y) ) ( Γ(x + y)b(x, y) y x π/ (sin θ) x (cos θ) y dθ ) Γ(x)Γ(x) Γ(x) B(x, x) t x ( t) x dt [t( t)] x dt / [t( t)] x dt (4t( t) s ) ( s 4 )x 4 ds x s x ( s) / ds s x x Γ(x)Γ(/) B(x, /) Γ(x + /) Γ(x)Γ(x + /) x Γ(/)Γ(x) π Γ(x) x

237 J n (x) (n ) J (), J n () (n ), J n() ( n ). n 3. [ ] n x (m+n) J m+n (x) ( ) n d {x m J m (x)} x dx [ ] n x m n d J m n (x) {x m J m (x)} x dx x d J dx + xdj dx + (x n )J (n ) J f(x)/ x f f + [ (n /4) x ] f x /x x J 4. e z (t t ) n J n (z)t n f(z, t) e z (t t ) w C f(z, w) πi C w dw n+ J n (z) π cos(z sin θ nθ)dθ π π 5. x, ϕ cos(x sin ϕ) J (x) + J m (x) cos(mϕ) m sin(x sin ϕ) J m+ (x) sin[(m + )ϕ] m 6. J / (x) cos x πx

238 n J n (x) J n (x) m J () n ( ) m ( ) x n+m (n ), Jn (x) ( ) n J n(x) (n < ) (n + m)!m! J (x) J n (x) J n () m m ( ) m m!m! ( x ) m + O(x ) ( ) m ( ) x n+m ( ) x n + O(x n+ ) (n + m)!m! n! n J n () ( ) n J n () J n () n J (x) + O(x ) J (x) O(x) J () n J n(x) n J n() m n n k J n(). 3.. ( ) m ( ) x n+m ( ) (n + m)!m! (n + m) n x n + O(x n+ ) n! J n (x) ( ) k J k (x) J n(x) ( ) k J k(x) d dx [xn J n (x)] x n J n (x), d dx [x n J n (x)] x n J n+ (x) d x dx [x m J m (x)] x (m+) J m+ (x) [ ] d [x m J m (x)] d x dx x dx [ x (m+) J m+ (x)] ( ) x (m+) J m+ (x) [ ]. n d [x m J m (x)] ( ) n x (m+n) J m+n(x) x dx

239 3 33 d x dx [xm J m (x)] x (m ) J m (x) [ ] d [x m J m (x)] d x dx x dx [x(m ) J m (x)] x (m ) J m (x) [ ] n d [x m J m (x)] x dx. x (m n) J m n(x) 3. J f x x / f d J dx df x / dx x 3/ f d dx J d f df x / x 3/ dx dx x 5/ f x d J dx + xdj dx + (x n )J [ x 3/ d f df x/ dx dx + 3 ] [ 4 x / / df f + x dx ] x / f + (x n )x / f d f df x dx dx x df f + x dx x f + (x n )x f x x d f dx + f d f dx + [ (n /4) x ] f A sin x + B cos x (A, B ) x J(x) (A sin x + B cos x) x 4..3 e z (t t ) n a n (z)t n, a n (z) πi C e z (w w ) dw w n+ w C w e iθ (θ π π) dw ie iθ dθ iwdθ a n (z) πi π C π π e z (w w ) dw w n+ πi iz sin θ e dθ e inθ π ( : θ π π π π π π e z (eiθ e iθ ) ie iθ dθ e i(n+)θ e i(z sin θ nθ) dθ π π π sin(z sin θ nθ)dθ ) cos(z sin θ nθ)dθ

240 3 34 e z (t t ) n J n (z)t n t n J n (z) π cos(z sin θ nθ)dθ π π 5. t e iϕ e x (t t ) n J n (x)t n e x (eiϕ e iϕ ) e ix sin ϕ cos(x sin ϕ) + i sin(x sin ϕ) n J n (x)(e iϕ ) n n J n (x)e inϕ J n (x)e inϕ n J n (x)[cos(nϕ) + i sin(nϕ)] n cos(x sin ϕ) J n (x) cos(nϕ) J n (x) cos(nϕ) + J (x) + J n (x) cos(nϕ) n n n J k (x) cos( kϕ) + J (x) + J n (x) cos(nϕ) k n ( ) k J k (x) cos(kϕ) + J (x) + J n (x) cos(nϕ) k n J (x) + J m (x) cos(mϕ) m 6. sin(x sin ϕ) J / (x) J n (x) sin(nϕ) J n (x) sin(nϕ) + + J n (x) sin(nϕ) n n n J k (x) sin( kϕ) + J n (x) sin(nϕ) k n ( ) k+ J k (x) sin(kϕ) + J n (x) sin(nϕ) J m+ (x) sin((m + )ϕ) k n m m m ( ) m ( ) x m / ( ) m ( x Γ(m + /)m! m Γ(m + /)Γ(m + ) ( ) m (m+/) ( ) x m / ( ) m πγ((m + )) x m πx m m! ) m / πx cos x

241 [ + a]ψ(x, y, z) (r, θ, ϕ) ϕ Φ(ϕ) d R Φ(ϕ) m ( ) Φ dϕ z Φ(ϕ) Φ(ϕ + π) Φ(ϕ) C ϕ e imϕ + C ϕ e imϕ (m ) θ Θ(θ) d Θ dθ + cos θ ( ) dθ sin θ dθ + λ m sin Θ (λ ) θ cos θ z Θ(θ) L(z) ( ) ( z ) d L dl z dz dz + λ m L z m ( z ) d L dl z dz dz + λl z L(z) n a n z n ( z ) a n n(n )z n z a n nz n + λ a n z n n n n a n+ (n + )(n + )z n a n n(n )z n a n nz n + λ a n z n n n n n (a + λa ) + (6a 3 a + λa )z + n[(n + )(n + )a n+ {n(n ) + n λ}a n ]z n a n+ n(n + ) λ (n + )(n + ) a n n a n+ a n a n n a n z n

242 33 36 n n(n + ) λ a n+ a n+4 a n+6 a n+, a n+3, a n+5,... n(n + ) λ n a n z n n a a a 3 a 5 n a a a a 4 L(z) P n (z) d n n n! dz n (z ) n dn dz n (fg) n d [(z n+ ) ddz ] dz n+ (z ) n k nc k ( dk f )(dn k g dz k n k ) dz (z ) dn+ dz n+ (z ) n + (n + )z dn+ dz n+ (z ) n + n(n + ) dn dz n (z ) n [ ] d [(z n+ ) ddz ] dz n+ (z ) n dn+ [ n(z ) n z ] n dn+ [ z(z ) n] dz n+ dz n+ [ ] n z dn+ dz n+ (z ) n + (n + ) dn dz n (z ) n [ ] n z dn+ dz n+ (z ) n + (n + ) dn dz n (z ) n (z ) dn+ dz n+ (z ) n + (n + )z dn+ dz n+ (z ) n + n(n + ) dn dz n (z ) n {n(n + ) n(n + )} dn dz n (z ) n (z ) dn+ dz n+ (z ) n + z dn+ dz n+ (z ) n n n! n(n + ) dn dz n (z ) n ( z ) dn+ dz n+ (z ) n z dn+ dz n+ (z ) n n(n + )P n (z) ( z ) d dz P n(z) z d dz P n(z) ( z ) d dz P n(z) z d dz P n(z) + n(n + )P n (z) P n (z)

243 33 37 P n (z) (z ) n n j z (n j) z n n j d n n! j!(n j)! z(n j) ( ) j dz n z(n j) (n j)(n j ) (n j n + )z n j (n j)! (n j)! (n j n)! zn j (n j)! zn j n j < P n (z) [n/] j ( ) j n! n n! j!(n j)! (n j)! (n j)! zn j [n/] n j ( ) j (n j)! j!(n j)!(n j)! zn j [x] x ( ) P n (z) P (z), P (z) z, P (z) (3z ), P 3 (z) (5z3 3z), cos θ z d n P n (x) n n! dx n (x ) n P n (x)p m (x)dx P n (x)p m (x)dx n n! m m! { dn dx n (x ) n }{ dm dx m (x ) m }dx [ ] { dn n+m n!m! dx n (x ) n }{ dm dx m (x ) m } n+m n!m! { dn dx n (x ) n }{ dm+ dx m+ (x ) m }dx (x ) n x n (x ) x ± P n (x)p m (x)dx n+m n!m! ( ) n+m n!m! ( ) n n+m n!m! { dn { dn dx n (x ) n }{ dm+ dx m+ (x ) m }dx dx n (x ) n }{ dm+ dx m+ (x ) m }dx (x ) n { dm+n dx m+n (x ) m }dx

244 33 38 n m { } x m n + m m n m n m P n (x)p n (x)dx ( )n (x ) n { dn n (n!) dx n (x ) n }dx ( )n n (n!) (n)! (x t ) ( )n n (n!) (n)! (4t 4t) n dt (n!) (n)! t n ( t) n dt (n)!b(n +, n + ) (n!) + )Γ(n + ) (n)!γ(n (n!) Γ(n + ) (n!) (n)! n!n! (n + )! n + (x ) n dx P n (x)p m (x)dx (n + ) δ mn 33.3 f(z) πi C f(w) w z dw z n (w, z C w z ) d n n! n f(z) dz πi C P n (x) d n n n! dx n (x ) n P n (x) P n (x)t n n t n n! n n! πi C n πi n (w ) n dw (w x) n+ C f(w) dw (w z) n+ n πi (w ) n t n dw (w x) n+ πi C C (w ) n (w x) (w x) n+ dw n (w )t (w x) < { (w } n [ ] )t (w )t (w x) (w x) { (w )t (w x) } n dw (w x) (w x) (w )t (w x)/t w (w x)/t w (w x)/t w t w + x t β w α, α t + t + x t t + t xt + t, β t t xt + t

245 33 39 t xt + t xt α /t, β x C β P n (x)t n (w x)/t n πi C (w x) [w (w x)/t] dw πit C (w α)(w β) dw πi [ ] πit Res (w α)(w β) t (β α) t wβ ( /t) xt + t xt + t 33.4 P n (x)t n n xt + t x t < P n ()t n n t + t ( t) t n P n () x x P n ( x)t n n ( x)t + t + x( t) + ( t) P n (x)( t) n P n (x)( ) n t n n n P n ( x) ( ) n P n (x) t [ ] P n (x)t n [ ] t n t xt + t P n (x)nt n n ( xt + t ) 3/ (t x) ( xt + t ) P n (x)nt n n ( xt + t ) / (t x) (x t) P n (x)t n n ( ) np n (x)t n x P n (x)nt n + P n (x)nt n+ n n n (n + )P n+ (x)t n x P n (x)nt n + P n (x)(n )t n n n n P (x) + [(n + )P n+ (x) nxp n (x) + (n )P n (x)] t n n ( ) xp (x) + xp n (x)t n P n (x)t n n n t n

246 33 4 P (x) x, P (x) P (x) xp (x) [(n + )P n+ (x) nxp n (x) + (n )P n (x) xp n (x) + P n (x)] t n n [(n + )P n+ (x) (n + )xp n (x) + np n (x)] t n n t (n + )P n+ (x) (n + )xp n (x) + np n (x) P (x), P (x) x n P n (x) ρ( r) r dv R k ρ( r)dv R r ( k ) 4πϵ R ϕ( R) V k ρ( r) R r dv R, r R, r θ R r ( R r) R + r Rr cos θ R (r/r) cos θ + (r/r) R r R cos θ(r/r) + (r/r) ( ) r n P n (cos θ) R n R ϕ( R) V k R ρ( r) k R k n R n+ V + k R 3 n V r ρ(r) ( ) r n ( ) k r n P n (cos θ) dv ρ( r)p n (cos θ) dv R n R V R ρ( r)p n (cos θ)r n r sin θdrdθdϕ ρ( r)r sin θdrdθdϕ + k R V V O ρ( r)r 3 cos θ sin θdrdθdϕ ρ( r) (3 cos θ ) r sin θdrdθdϕ + P n n R r O 3 R

247 ( z ) d L dl z dz dz + ( m n(n + ) z ) L m ( λ n(n + ) ) m m > z ± z z L(z) ( z) s f(z) ( z )[s(s )( z) s f s( z) s f + ( z) s f ] ( ) z[ s( z) s f + ( z) s f ] + z t t m n(n + ) z t( t)[s(s )t s f st s f + t s f ] ( t)[ st s f + t s f ] + ( ( z) s f n(n + ) ) m t s f t( t) t s ( ) ( t) s(s )f s( t)f + t( t)f ( t) + s f ( t)f + n(n + ) m f t t t( t) [ ] (/t ) s(s ) + s m f + O() t t /t s(s ) + s m s ±m z L s m/ z ( z z ) L(z) ( z) m/ ( + z) m/ f(z) ( z ) m/ f(z) ( z ) d f df (m + )z + {n(n + ) m(m + )}f dz dz m P n (z) z ( z ) d P n dz z dp n dz + n(n + )P n ( z ) d3 P n dz 3 ( z ) d3 P n dz 3 z d P n dz 4z d P n dz z d P n dz dp n dz + n(n + )dp n dz + {n(n + ) }dp n dz

248 33 4 dp n m dz f ( z ) d f df (m + )z + {n(n + ) m(m + )}f dz dz z ( z ) d3 f dz 3 z d f dz (m + )z d f df (m + ) dz dz ( z ) d3 f dz 3 (m + )z d f dz + {n(n + ) m(m + )} df dz df + {n(n + ) (m + )(m + )} dz df dm m f(z) dz dz m P n(z) L(z) ( z ) m/ f(z) ( z m/ dm ) dz m P n(z) Pn m (z) Pn m (z) ( ) P n (z) z n m n m P n (z) d n n n! dz n (z ) n Pn m (z) n n! ( z m/ dn+m ) dz n+m (z ) n m n m n, n +,..., n, n 33.6 θ, ϕ Y (θ, ϕ) [ Y θ + cos θ Y sin θ θ + ] Y sin θ ϕ λy (λ ) ϕ e ±imϕ (m ) Y (θ, ϕ) e ±imϕ L(θ) [ e ±imϕ L θ + cos θ ] L sin θ θ m sin θ L λe ±imϕ L L θ + cos θ ( ) L sin θ θ + λ m sin L θ cos θ z ( ( z ) d L dl z dz dz + λ L(z) P m n (z) C m n π π Yn m n m z Y m n (θ, ϕ) C m n e imϕ P m n (cos θ) n, m ) L {Y m n (θ, ϕ)} Y m n (θ, ϕ) sin θdθdϕ δ nn δ mm n m n

249 z cos θ d dθ + cos θ d sin θ dθ ( z ) d dz z d dz k(k + ) n(n + ). {a n } a k+ a k (n ) (k + )(k + ) P n (z) a k z k k (a) n, a, a (b) n, a /, a (c) n 3, a, a 3/ 3. P n (z) d n n n! dz n (z ) n n,,, 3 4. P n ( z) ( ) n P n (z) P n+ () 5. n < m dn dx n (x ) m x ± 6. n, m,, P n (x)p m (x)dx 7. P (), P 4 (), P 6 () 8. P n () ( )n (n)! n (n!) (n + ) δ mn 9. P n (x)t n n xt + t x P n+(x) xp n(x) + P n (x) P n (x). (a) P n+(x) xp n(x) (n + )P n (x) (b) xp n(x) P n (x) np n (x) (c) P n+(x) P n (x) (n + )P n (x). (c) (n + )P n+ (x) (n + )xp n (x) + np n (x) P n (x)dx (n + ) [P n () P n+ ()]

250 L(z) ( z ) m/ f(z) f ( z ) d f df (m + )z + {n(n + ) m(m + )}f dz dz 3. P, P, P, P, P, P, P, P, P 4. Y m n (π θ, ϕ + π) ( ) n Y m n (θ, ϕ) 5. Y m n (θ, ϕ) C m n e imϕ P m n (cos θ) Y, Y, Y

251 z cos θ d dθ d dθ d dθ dz dθ d dz sin θ d ( ) dz sin θ d dz z d dz + ( z ) d dz d dθ + cos θ d sin θ dθ d dθ + cos θ d sin θ dθ cos θ d dz dz d sin θ dθ dz z d dz + sin θ d dz z d dz + ( z ) d dz z d dz ( z ) d dz z d dz. a k+ k(k + ) n(n + ) a k (k + )(k + ) ( + ) ( + ) (a) n k a 3 a + a 3 ( + ) ( + ) n a n a n a n a P (z) a z z ( + ) ( + ) (b) n k a 4 a + a 4 ( + ) ( + ) n a n a n a n a a ( + ) ( + ) a 3 ( + )( + ) P (z) a z + a 3 z (3z ) 3 (3 + ) 3 (3 + ) (c) n 3 k 3 a 5 a 3+ a 3 5 (3 + ) (3 + ) n a n a n a n a 3 a 3 ( + ) 3 (3 + ) a 5 ( + )( + ) P 3 (z) a 3 z 3 + a z 5 z3 3 z (5z3 3z)

252 P (z)! (z ) P (z) d! dz (z ) z z P (z) d! dz [(z ) ] d 8 dz [(z )z] (z 4) (3z ) 8 P 3 (z) d 3 3 3! dz 3 [(z ) 3 ] d 48 dz [6z(z ) ] d 48 dz [6z5 z 3 + 6z] d 8 dz [5z4 6z + ] 8 (z3 z) (5z3 3z) 4. P n (z) d n n n! dz n (z ) n P n ( z) d n n n! d( z) n [( z) ] n ( ) n d n n n! dz n (z ) n ( ) n P n (z) P n+ () P n+ ( ) ( ) n+ P n+ () P n+ () P n+ () 5. (x ) m (x ) m x n m > n d dx (x ) (x ) x ± 6. P (x), P (x) x, P (x) (3x ) P (x)p (x)dx P (x)p (x)dx P (x)p (x)dx P (x)p (x)dx P (x)p (x)dx P (x)p (x)dx dx x dx + ( + ) 4 (9x4 6x + ) dx ( ) 5 ( + ) x dx (3x ) dx x (3x ) ( ) dx (3x 3 x) ( ) 7. (n + )P n+ (x) (n + )xp n (x) + np n (x) P n+ (x) (n + ) (n + ) xp n(x) n (n + ) P n (x)

253 33 47 P () P () P () 3 P () P () P 4 () 7 4 P 3() 3 4 P () 3 8 P 6 () 6 P 5() 5 6 P 3() P n (z) P n (z) d n n (n)! dz n (z ) n (z ) n n k nc k (z ) k ( ) n k n k nc k z k ( ) k z n z k n z n d n dz n [ nc n z n ( ) n ] ( ) n nc n (n)! ( ) n (n)!(n)! n!n! P n () d n n (n)! dz n (z ) n z (n)!(n)! n ( )n (n)! n!n! ( )n (n)! n (n!) 9. x [ ] P n (x)t n x n x [ ] xt + t P n(x)t n n ( xt + t ) 3/ ( t) ( xt + t ) P n(x)t n n ( xt + t ) / ( t) t P n (x)t n n ( ) P n(x)t n x P n(x)t n+ + P n(x)t n+ n n n P (x) + P (x)t + P n+(x)t n+ xp (x)t n x P n(x)t n+ + P n (x)t n+ n n [ t + P n+ (x) xp n (x) + P n (x) ] t n+ n ( ) t + P n (x)t n n t n+ n P n+(x) xp n(x) + P n (x) P n (x)

254 (n + )P n+ (x) (n + )xp n (x) + np n (x), P n+(x) xp n(x) + P n (x) P n (x) (a) (n + )P n+ (x) (n + )xp n (x) + np n (x) x (b) 9 (n + )P n+(x) (n + )P n (x) (n + )xp n(x) + np n (x) n [ P n+(x) xp n(x) + P n (x) ] (n + )P n (x) + P n+(x) xp n(x) 9 np n (x) (n + )P n (x) + P n+(x) xp n(x) P n+(x) xp n(x) (n + )P n (x) P n+(x) xp n(x) + P n (x) P n (x) P n+(x) xp n(x) [ xp n(x) P n (x) ] P n (x) ( (a) ) (n + )P n (x) [ xp n(x) P n (x) ] P n (x) (c) (a) (b) xp n(x) P n (x) np n (x) [ P n+ (x) xp n(x) ] + [ xp n(x) P n (x) ] (n + )P n (x) + np n (x) P n+(x) P n (x) (n + )P n (x). (n + )P n (x) P n+(x) P n (x) x (n + )P n (x) dx (n + ) [ P n+ (x) P n (x) ] dx P n (x) dx [P n+ (x) P n (x)] ( n P n () ) (n + ) P n (x) dx [P n+ () P n ()] P n (x) dx (n + ) [P n () P n+ ()]

255 ( ( z ) d L dl z dz dz + n(n + ) L(z) ( z ) m/ f(z) m z ) L d dz L m ( z ) ( m ) ( z)f + ( z ) m f mz( z ) ( m ) f + ( z ) m f d dz L m( z ) ( m ) f + m(m )z ( z ) ( m ) f mz( z ) ( m ) f + ( z ) m f ( z ) [ m( z ) ( m ) f + m(m )z ( z ) ( m ) f mz( z ) ( m ) f + ( z ) m f ] z [ mz( z ) ( m ) f + ( z ) ] m f + n(n + ) ( z ) m z f [ ( z ) m m(m )z mf + f mzf + ( z )f z ( ) ] + mz z f zf + n(n + ) m f z [ ( z ) m ( z )f (m + )zf + {n(n + ) (m + m)}f ] [ ( z ) m ( z )f (m + )zf + {n(n + ) m(m + )}f ] ( z )f (m + )zf + {n(n + ) m(m + )}f ( m ) 3. P (z) (z ) Pn m (z) n n! ( z m/ dn+m ) dz n+m (z ) n P (z) ( z / d ) dz (z ) ( z ) / d dz (z) ( z ) / P (z) d dz (z ) z P (z) ( z ) / (z ) ( z ) / P (z)! ( z ) d4 dz 4 (z ) 8 ( z ) d4 dz 4 (z4 z + ) 3( z ) P (z)! ( z ) / d3 dz 3 (z ) 8 ( z ) / d3 dz 3 (z4 z + )

256 ( z ) / (4z) 3z( z ) / P (z) d! dz (z ) d 8 dz (z4 z + ) 8 (z 4) (3z ) P (z)! ( z ) / d dz (z ) 8 ( z ) / (z )(z) z( z ) / P (z)! ( z ) (z ) 8 ( z ) 4. Y m n (θ, ϕ) C m n e imϕ P m n (cos θ) Y m n (π θ, ϕ + π) C m n e im(ϕ+π) P m n (cos(π θ)) ( ) m C m n e imϕ P m n ( cos θ) Pn m (z) n n! ( z m/ dn+m ) dz n+m (z ) n 5. 3 P m n ( z) n n! ( ( z) ) m/ d n+m d( z) n+m (( z) ) n ( ) n+m P m n (z) Yn m (π θ, ϕ + π) ( ) m Cn m e imϕ ( ) n+m Pn m (cos θ) ( ) n Yn m Y C e iϕ P (cos θ) C e iϕ sin θ, Y C P (cos θ) C cos θ Y C e iϕ P (cos θ) C π π e iϕ ( ) sin θ C e iϕ sin θ {Y m n (θ, ϕ)} Y m n (θ, ϕ) sin θdθdϕ δ nn δ mm C, C, C π π {Y (θ, ϕ)} Y (θ, ϕ) sin θdθdϕ C π (cos θ z ) C π z ( )dz C 4π( 3 ) C 8π 3 π π π {Y (θ, ϕ)} Y (θ, ϕ) sin θdθdϕ C C π π π C 4 z dz C 4π 3 C 4π 3 {Y (θ, ϕ)} Y π (θ, ϕ) sin θdθdϕ C 4 8π 3 ( z )dz C 4 π π π π sin θ sin θdθdϕ cos θ sin θdθdϕ sin θ sin θdθdϕ C m n 3 3 C 8π, C 4π, 3 Y ± ± 8π e±iϕ sin θ, Y ± C 3 π 3 4π cos θ

257 5 cosh, 5 e, 4 ln ( ), 6 MKSA, 4 order estimation, 4 sinh, 5 SI, 4 tanh, 5, 74, 58,,, 9, 8, 5, 75, 9,, 48, 74, 53 (rot), 9, 7, 64, 7, 58,, 87, 9, 7, 39, 4, 64, 6, 9, 8, 4, 47, 89, 9, 3, 37, 39, 5, 38,, 38, 4,, 6, 47, 46, 8 i, 8, 8, 67, 66, 66 δ, 5, 38, 5, 88, 3 (grad), 9, 46, 36, 9, 6, 6, 9, 45,,, 5, 5, 4, 46, 9, 9, 6, 4, 8, 8, 35, 63, 65, 43, 8,,, 7, 86, 7, 8, 6, 4, 43, 3, 7, 68, 6, 5, 58, 3, 3, 5, 46,, 4, 4, 9

258 5, 38, 4, 3, 8, 88, 53,, 5 δ, 5, 74, 86,, 7, 9, 8, 7, 9,, 89, 74, 4, 9,, 5, (div), 9, 64, 85,, 5, 8, 74, 69, 63, 3, 66,, 34, 9, 8, 64, 37, 38, 37, 4, 35, 6, 47, 43, 3, 43, 8, 8, 57, 59,, 8, 8, 9, 8, 35, 9, 5, 5, 9, 4, 3, 86, 86, 3, 86, 3, 94, 5, 8, 4, 64, 9, 8, 8, 6, 8, 64, 8

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熊本県数学問題正解

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