Transcription

1 kawahiraamath.titech.ac.jp

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3 ii 5 50 () : (2) R: (3) Q: (4) Z: (5) N: (6) : () α: (2) β: (3) γ, Γ: (4) δ, : (5) ϵ: (6) ζ: (7) η: (8) θ, Θ: (9) ι: (0) κ: () λ, Λ: (2) µ: (3) ν: (4) ξ, Ξ: (5) o: (6) π, Π: (7) ρ: (8) σ, Σ: (9) τ: (20) υ, Υ: (2) ϕ, Φ: (22) χ: (23) ψ, Ψ: (24) ω, Ω: () x X x X x X (2) X {x X x } N = {n Z n > 0} (3) X Y X Y ( (4) A := B A B e := lim + n. n n)

4 iii (5) : ( 2) 2 {a n } : M n a n M

5 i 2. i = iv

6 v

7

8 . i = 2. x 2 = 2 ± ( + 2)(3 2) = ( 2) 2 2 = x 2 = 2 ( + )(3 ) = ( ) 2 = i a b a + bi i (complex number) B B (9 ) UFO 2

9 .2. 3 := {a + bi a, b R}. b = 0 a + 0i a R 3 z = a + bi, w = c + di (0) a + bi = 0 : a = b = 0 ( a 2 + b 2 = 0) () z ± w := (a ± c) + (b ± d)i (2) zw := (ac bd) + (ad + bc)i (3) w 0 z w ac + bd ad + bc := + c 2 + d2 c 2 + d i 2. (0) () a + bi = c + di (a c) + (b d)i = 0 a = c b = d z z = a + bi a + bi.2 (0) (3) xy 2 R 2 R 2 := {(a, b) a, b R}. := {a + bi a, b R}. 3 (0)

10 .2. 4 (a, b) a + bi xy R 2 (a, b) a + bi 2 i = 0 + i (0, ) R 2 (0) (3) 2 (0) () (0) (a, b) = 0 a = b = 0 () (a, b) ± (c, d) = (a ± c, b ± d) (2) (3) (2) (2) (a, b) (c, d) := (ac bd, ad + bc) 4 i 2 (0, ) (0, ) = (, 0) 4 (2) b = 0 (a, 0) (c, d) := (ac, ad)

11 .2. 5 i (3) (0) (3) xy 5 xy R 2 (a, b) (a, 0) (0, b) (0, 0) x y a + bi a R bi Ri 0 R Ri. z = a + bi a z (real part) b z (imaginary part) a = Re z, b = Im z x y 6 z = a + bi a bi z (complex conjugate of z) z (5) (6) - z = a + bi, w () (z) = z (2) z + w = z + w (3) zw = z w (4) (5) Re z = z + z 2 (6) Im z = z z 2i (7) zz = a 2 + b 2 ( 0) ( z ) = z w w 5 complex plane pure imaginary number real axis, imaginary axis. 6 Im z = b R Im z = bi

12 z = a + bi 0 xy z (absolute value) (modulus) z - z = a 2 + b 2 = zz. z 0 0 z (argument) arg z 78 (radian) i =, arg i = π/2, + i = 2, arg( + i) = π/4. z = a + bi 0 r = z > 0 arg z = θ a = r cos θ, b = r sin θ z z = r(cos θ + i sin θ) z (polar form) (polar representation) 9 = cos 0 + i sin 0 i = cos π 2 + i sin π 2 = cos π + i sin π i = cos 3π 2 + i sin 3π 2 + i = 2(cos π 4 + i sin π 4 ) 3 + i = 2(cos 5π 6 + i sin 5π 6 ) 7 arg 0 8 2π arg i = π/2 arg i = 5π/2 arg i = 3π/2 0 arg z < 2π 9 r = z = cos θ + i sin θ z = e iθ e iθ = cos θ + i sin θ

13 z = a + bi iz = i(a + bi) = b + ai π/2 i(iz) = z = a bi π/2 i π/2 = i 2 π z w z = r(cos θ + i sin θ), w = r (cos θ + i sin θ ). (2) zw = rr (cos θ + i sin θ)(cos θ + i sin θ ) = rr {(cos θ cos θ sin θ sin θ ) + i(sin θ cos θ + cos θ sin θ )} = rr {cos(θ + θ ) + i sin(θ + θ )} w = i r =, θ = π/2 iz = r{cos(θ + π/2) + i sin(θ + π/2)} i π/2

14 .3. 8 w = r (cos θ +i sin θ ) r θ 0,, z w w arg w 0, w, zw -2. z, w 0 () zw = z w arg zw = arg z + arg w (2) z = w z w arg z w = arg z arg w ()

15 2 2. z = r(cos θ + i sin θ), w = r (cos θ + i sin θ ) zw = rr {cos(θ + θ ) + i sin(θ + θ )}. z = w 2 z 2 = r 2 (cos 2θ + i sin 2θ) ( ) z = r(cos θ + i sin θ) m z m = r m (cos mθ + i sin mθ). A = ( + 3 i) i 2(cos π 3 + i sin π 3 ) A = 2 0 (cos 0π 3 + i sin 0π 3 ) = 024(cos 4π 3 + i sin 4π 3 ) = i. 2.2 x e x = + x! + x2 2! + x3 3! + 9

16 x iθ (θ R) e iθ = +(θi)+ (θi)2 2! + (θi)3 3! i+ (θi)4 + = 4! ) ) ( θ2 2! + θ4 4! +i (θ θ3 3! + θ5 5! cos x = x2 2! + x4 x3 sin x = x 4! 3! + x5 5! θ e iθ = cos θ + i sin θ. e e iθ θ 2.3 z = x + yi (x, y R) e z := e x (cos y + i sin y) z (exponential function) e z exp z 2 e 99 = + 99! ! ! +

17 2.4. e z e x > 0 y. e x > 0 e z 0. e z e z. z = 0 = 0 + 0i e 0 := e 0 (cos 0 + i sin 0) =.. z = πi = 0 + πi e πi := e 0 (cos π + i sin π) =. e, π, i. z = 3 + π ( 4 i e3+πi/4 := e 3 cos π 4 + i sin π ) ( = e ) i z = r > 0, arg z = θ z z = r(cos θ + i sin θ) z = re iθ z = re iθ z + i = ( 2 cos π 4 + i sin π ) = 2e πi/4 4 2 = 2(cos π + i sin π) = 2e πi 2.5

18 ( ). z, w () e z e w = e z+w (2) ez e w = ez w (3) e z+2πi = e z 2-. () z = x + yi, w = x + y i z + w = (x + x ) + i(y + y ) e z+w := e x+x {cos(y + y ) + i sin(y + y )}. e z e w = e x (cos y + i sin y) e x (cos y + i sin y ) = e x+x {cos(y + y ) + i sin(y + y )}. (2) () e z = e (z w)+w = e z w e w e z /e w = e z w. (3) e 2πi := cos 2π + i sin 2π = () e z+2πi = e z e 2πi = e z. () (2) (3) (3) = e z 4πi = e z 2πi = e z = e z+2πi = e z+4πi = 2πi z = 0 m = e 0 = e 2πmi () (e z ) n = e z e z = e z+ +z = e nz. e z e z = e z+( z) = e 0 = e z = /e z = (e z ) e z e z 2-2 m Z (e z ) m = e mz. z = r(cos θ + i sin θ) m z m = (re iθ ) m = r m e imθ = r m (cos mθ + i sin mθ) N. N z N = N w z N = w w N

19 z 5 = z = re iθ (r > 0, 0 θ < 2π) ( =) z 5 = r 5 e 5θi r 5 = r > 0 r = = e 5θi = e 2mπi (m Z) ( ) 5θ = 2mπ (m Z). 0 θ < 2π θ = 2mπ 5 (m = 0,, 2, 3, 4) 5 = e 0, e 2πi/5, e 4πi/5, e 6πi/5, e 8πi/ N N N z N = ( ) 2mπi z = exp (m = 0,,..., N ) N N N

20 3 3. z = x + yi (x, y R) e z e x > 0 y. := e x (cos y + i sin y) z, w e z e w = e z+w z, w e z+2πi = e z 3.2 w = f(z) = e z f 2 2 =4 f : z- w- f z w 4

21 z z- z = x + yi (x, y R). x = a z = a + yi (y R) w = e z = e a (cos y + i sin y) e a > 0 y z x = a w e a z 2π 2πi w a. y = b z = x + bi (x R) w = e z = e x (cos b + i sin b) b e x z y = b w b 3.3 a e x = a log a a

22 α 0 e z = α α (logarithm) log α. log( ) e z = = = e πi = e πi+2mπi (m Z) e z z = (2m + )πi (m Z). log( ) = (2m + )πi (m Z). log( + i) e z = + i e z = + i = 2 e πi/4 = e log 2 e πi/4+2mπi = e log 2+(π/4+2mπ)i (m Z) z = (log 2)/2 + (π/4 + 2mπ)i (m Z). log( + i) = log ( π 4 + 2mπ ) i (m Z). α = e z = = e 2mπi (m Z) z = 2mπi (m Z). log = 2mπi (m Z) log log = 0 2πi 2πi log α e z = α x 3 = ω ω

23 z = re iθ 0 (r > 0, θ R) r = e log r log z = log r + (θ + 2mπ)i (m Z) = log z + (arg z)i log arg z +2mπ (m Z) 0 z log z (logarithmic function) A A 2 log z 2πi 0 Im (log z) < 2π log z (principal value) Log z 3 z Log z Log ( ) = πi, Log ( + i) = log π i, Log = z 0 α z α (z to the power of α) z α := e α log z log z z α 2 (multivalued function) (single-valued function) 3 π Im (Log z) < π Log z log z

24 ( ). ( ) i ( ) i := e i log( ) = e i(2m+)πi (m Z) = e (2m+)π (m Z)...., e 3π, e π, e π, e 3π, e 5π,... 2( ). ( + i) 2 = ( + i)( + i) = 2i ( + i) 2 := e 2 log(+i) log 2 = e 2 { 2 + ( π 4 +2mπ )i} (m Z) = e log 2+ ( π 2 +4mπ )i (m Z) = e log 2 e πi/2 = 2 i = ( + i)( + i). 3( ). /3 = 3 = /3 = e (/3) log (log ) = e (/3) 2mπi = e 2mπi/3 (m Z) =, e 2πi/3, e 4πi/3 3 z 0 z /N (N N) z N. e = z e z. (e z ) w = e zw (e z ) w e z w e zw zw z = πi, w = i = (e πi ) i = ( ) i e πi i = e π

25 4 4. θ R e iθ = cos θ + i sin θ θ θ e iθ = cos θ i sin θ cos θ = eiθ + e iθ, sin θ = eiθ e iθ. 2 2i θ z z cos z := eiz + e iz, sin z := eiz e iz 2 2i (trigonometric function). cos i = ei i + e i i 2 = 2 ( ), ei i e + e e i i sin i = 2i = 2 ( e + e ) i. sin i / R cos i > e/2 > cos z, sin z 4-0 z, w () cos z = cos(z + 2π), sin z = sin(z + 2π) (2) cos 2 z + sin 2 z = 9

26 (3) cos(z ± w) = cos z cos w sin z sin w sin(z ± w) = sin z cos w ± cos z sin w 4-2. x 3 = 0 x = x =, e 2πi/3, e 4πi/3 sin z = 0 eiz e iz = 0 e iz = e iz 2i (e iz ) 2 = e 2iz = = e 2mπi (m Z). 2iz = 2mπi z = mπ (m Z) cos z z α, β (a) z (z α ) = z α 0 z α (b) z α f(z) β f(z) β (z α) lim z α f(z) = β β f(z) z α (limit) (a) z α 0 R 2 4- z = x + yi, α = a + bi (x, y, a, b R) z α x a y b.

27 R (triangle inequality) 4. z, w z w z + w z + w. 4-4.: z α = x a 2 + y b 2 x a 2 = x a z α = x a z α = y b z α = (x a) + i(y b) x a + y b. x a y b z α lim z α f(z) = A, lim z α g(z) = B () lim z α {f(z) + g(z)} = A + B (3) B 0 lim z α f(z) g(z) = A B (2) lim z α f(z) g(z) = A B

28 w = f(z) z = α (continuous) z α f(z) f(α), lim z α f(z) = f(α) D f D f D f(z) z = α f (continuous function) 2. f(z) = z 2 α z α f(z) f(α) = z 2 α 2 = (z + α)(z α) ( 4 2) 2α 0 = 0 z = α α f(z) = z 2. exp(z) = e z α z := z α z =: x + i y ( x, y R) e z e α = e α (e z α ) = e α (e z ) = e α {e x (cos y + i sin y) } z α z 0 x, y 0 4- e z α ( 4 2) e e α {e 0 (cos 0 + i sin 0) } = 0 z = α OK α exp(z) = e z.

29 5. w = f(z) = z 2 f z = i w = z = i f w = i z z := z i f(i) f(z) w := f(z) f(i) w = f(z) f(i) = (i + z) 2 ( ) = 2i z + z 2 z 0 0 z 2 w 2i z 2i = 2e πi/2 w z 2 90 z 2 z 23

30 24. w = f(z) z = α z = z α w = f(z) f(α) w A z z 0 z A w = f(z) z = α (differentiable) : A f(z) f(α) lim z α z α = A A f α (differential coefficient) A = f (α) z α f(z) f(α) z α A w z A w A z A = re iθ 0 f z = α r = A θ = arg A f(α)

31 25. f(z) = z 2 z = i f (i) = 2i. 2 f(z) = z 2 α f (α) = 2α f(z) = z 2 α f(z) f(α) z α = z2 α 2 z α = z + α 2α (z α) z = α f f (α) 2α α f. z α z α 0 f(z) = z 2, α = +i f(α) = 2i, f (α) = 2(+i) = 2 2e πi/4 f + i 2i 2 2 π/4 z α = + i f (α) = 2 2e πi/4 f(z) f(α) g(z) = z w = f(z) z = α f(z) f(α) z α A z α z α f(z) f(α) A z α z α f(z) f(α) ( A + ) z α z = α

32 D w = f(z) D : α D f D w = f(z) z D f (z) f (derivative) w = f(z) D (holomorphic) : f D f D. f(z) = z 2 f (z) = 2z f. e z sin z, cos z. 5.2

33 g(z) = z α. α z = z α = re iθ g(z) g(α) z α = z α z α = re iθ re iθ = e 2θi. z α z = re iθ 0 θ θ = 0 θ = π/2 z α z = α θ g θ 2θ θ

34 6 6. f(z) f(α) w = f(z) z = α : A, lim = A z α z α A A = f (α) z f (z) f w = f(z) D : 6.2 z = α D { f f 6- f(z), g(z) (DF) {f(z) + g(z)} = f (z) + g (z) (DF2) {f(z)g(z)} = f (z)g(z) + f(z)g (z) { } f(z) (DF3) g(z) 0 = f (z)g(z) f(z)g (z) g(z) {g(z)} 2 (DF4) {g(f(z))} = g (f(z)) f (z). 28

35 (DF4) α f g f(α) g(f(α)) f f (α) g g (f(α)) z = x + yi 2 (x, y) R 2 f : z w = u + vi (u, v R) F : x u u(x, y) = y v v(x, y) f F 2. f : w = f(z) = z 2 f(x + yi) = (x 2 y 2 ) + 2xyi f F : x u = x2 y 2 y v 2xy F : R 2 R 2. f : w = f(z) = e z f(x + yi) = e x (cos y + i sin y) f F : x u = ex cos y y v e x sin y

36 F : R 2 R 2. g : w = g(z) = z + 2z f(x + yi) = (x + yi) + 2(x yi) = 3x yi f F : x u = 3x y v y F : R 2 R 2 F : x u u(x, y) = f y v v(x, y) u = u(x, y) ( -function) (x, y) u x (x, y), u y (x, y). u = u(x, y) u (x, y) = (a, b) x := x a, y := y b, u := u(x, y) u(a, b) x, y 0 u P x + Q y P = u x (a, b) Q = u y (a, b) u = u(x, y) u(x, y) = u(a, b) + u x (a, b) (x a) + u y (a, b) (y a) +

37 u = P x + Q y + R( x, y); R( x, y) ( x, y) (0, 0) 0 x2 + y 2 R( x, y). F : x y u u = v u(x, y), v = v(u, v) F (a, b) u = P x + Q y + R ( x, y) = v P 2 x + Q 2 y + R 2 ( x, y) = u v P Q x + R ( x, y) P 2 Q 2 y R 2 ( x, y) P Q x ( ) P 2 Q 2 y P Q = u x(a, b) u y (a, b) P 2 Q 2 v x (a, b) v y (a, b) (Jacobian matrix) ( x, y) (0, 0) R k( x, y) 0 x2 + y 2 k =, 2 ( ) F : x u a y v b F (a, b) P Q P 2 Q 2

38 F : x u u(x, y) = f y v v(x, y) f P Q P 2 Q 2 = r cos θ sin θ sin θ cos θ : r θ u x(a, b) u y (a, b) = r cos θ r sin θ v x (a, b) v y (a, b) r sin θ r cos θ ( ) u x = v y (= r cos θ) v x = u y (= r sin θ) α := a + bi f r θ f (α) = re iθ = r(cos θ + i sin θ) = u x + iv x u, v f (x+yi) z = x+yi 6-2. D w = f(z) f(x + yi) = u + vi (x, y, u, v R) (a) f D u (b) u, v D x = v y (R) v x = u y f (z) = u x + iv x (R) auchy-riemann equation)

39 w = f(z) = z 2 F : x u = x2 y 2 y v 2xy x u y = 2x 2y 2y 2x u, v (R) f (z) = 2z = 2x + 2yi = u x + iv x. w = f(z) = e z f (z) = e z F : x u = ex cos y y v e x sin y u x u y = ex cos y e x sin y u, v e x sin y e x cos y v x v x v y (R) x, y 6-2 f f (z) = u x +iv x = e x cos y+ie x sin y = e z v y 6-3 (e z ) = e z (sin z) = cos z (cos z) = sin z cos z = (e iz + e iz )/2. w = g(z) = z + 2z F : x y u = 3x = 3 0 x u x u y = v y 0 y v x v y 3 0 u, v (R) f g 3

40 F : x u = x2 + y 2 y v 2xy f(z) = z 2 x u y = 2x 2y u, v 2y 2x v x v y (R) 6-2 f f(z) = z (a) = (b). f : D α = a + bi D z = x + i y = z α, w = u + i v = f(z) f(α), f (α) = P + iq x, y, u, v, P, Q R w z f (α) ( z 0) γ( z) := w f (α) z = (f(z) f(α)) f (α)(z α) w = f (α) z + γ( z), γ( z) z 0 ( z 0) u + i v = (P + iq)( x + i y) + γ( x + i y) u + i v = (P x Q y) + i(q x + P y) + γ( x + i y) ( ) ( ) ( ) ( ) u P Q x Re γ( x + i y) = + v Q P y Im γ( x + i y) u(x, y), v(x, y) (x, y) = (a, b) u(a + x, b) u(a, b) x P ( x 0)

41 u = P x Q y + Re γ( x + i y) y = 0, x 0 u(a + x, b) u(a, b) x = u x = P + Re γ( x + i 0) x + i 0 z = x + i 0 0 = Re γ( z) z γ( z) z 0. u x (a, b) = P (R) f (α) = P + iq = u x + iv x a + bi f (a + bi) = P + Qi a, b u, v (b) (b) = (a). (b) P := u x (a, b) = v y (a, b) Q := v x (a, b) = u y (a, b) u, v ( ) ( ) ( ) ( ) u P Q x R ( x, y) = + v Q P y R 2 ( x, y) 2 u + i v = (P + Qi)( x + i y) + {R ( x, y) + ir 2 ( x, y)} w = (P + Qi) z + γ( z). γ( z) = γ( x + i y) := R ( x, y) + ir 2 ( x, y) ( x, y) (0, 0) z 0 γ( z) z = R ( x, y) + ir 2 ( x, y) x 2 + y 2 R (x, y) + R 2 (x, y) x 2 + y 2 0. f z = α f (α) = P + Qi = u x (a, b) + v x (a, b)i u, v α = a + bi α f D

42 D w = f(z) f(x + yi) = u + vi (x, y, u, v R) (a) f D (b) u, v D (R) f (z) = u x + iv x { ux = v y v x = u y (R) auchy-riemann equation) 7.2 x, y, u, v R. f(x+yi) = u+vi u = u(x, y) u = y 3 3x 2 y v = v(x, y). f(x + yi) = u + vi u = u(x, y) f. f(x + yi) = u + vi u = u(x, y) = x 2 + y 2 f. (R) u x = 6xy = v y () u y = 3y 2 3x 2 = v x (2) () v y v = 3xy 2 + g(x) 36

43 g(x) x x v x = 3y 2 + d dx g(x) (2) d dx g(x) = 3x2 g(x) = x 3 + v v = 3xy 2 + x 3 + ( R) v u, v (R) f. f(z) = i(z 3 + ) ( R). u (R) u x = 0 = v y u y = 0 = v x v = ( R) f = u + vi. f (R) u x = 2x = v y () u y = 2y = v x (2) () v = 2xy + g(x) v x = 2y (2) f(z) if(z)

44 y = f(x) [a, b] b a f(x) dx y = f(x) y = 0 [a, b] a = x 0 < x < < x N = b N f(x k )(x k+ x k ) k=0 N x k+ x k 0 a b α β

45 (curve) x = x(t), y = y(t) R = {z(t) = x(t) + iy(t) t [a, b]} t (orientation) z(a) (initial point) z(b) (terminal point) a b. R 2 z(t) = x(t) + iy(t) (x(t), y(t)) R 2 x, y t d x(t) = x (t) R 2 dt y(t) y (t). : z = z(t) = x(t) + iy(t) (t [a, b]) (partition) a = t 0 < t < < t N < t N = b {t k } N k=0 {z(t k )} N k=0 z k := z(t k ) t k

46 D f : D D := {z k = z(t k )} N k=0 f (Riemann sum) Σ(f, ) Σ(f, ) := N f(ζ k ) (z k z k ) k= ζ k k t k s k t k s k ζ k := z(s k ) f(ζ k ) (z k z k ) 7.: I max z k+ z k {ζ, ζ 2,..., ζ N } 0 k N I Σ(f, ) 0 I f(z) dz f z(t) = t (t [a, b]) f : [a, b] R f(z) dz = b a f(t) dt R

47 7.5. 4

48 D f : D : z = z(t) = x(t) + iy(t) (a t b) f(z) dz = b z (t) = d z(t) dt a f(z(t)) z (t) dt z (t) = dz (t) z(t) dt t dz (t) := lim dt t 0 dz dx (t) = dt dt z(t + t) z(t) t (t) + idy (t) dt xy. f(z) = z 2 : z = z(t) = t + ti (0 t ) z (t) = + i z 2 dz = (t + ti) 2 ( + i) dt = ( + i) 3 t 2 dt = 0 0 ( + i)

49 z()3 3 z(0)3 3. α r > 0 (α, r) := { z(t) = α + re it 0 t 2π } t 8-2 m Z, = (α, r) r > 0 0 (m ) (z α) m dz = 2πi (m = ) 8-2 z(t) = α + re it z (t) = ire it. (z α) m dz = 2π 0 2π (re it ) m (ire it ) dt = ir m+ e i(m+)t dt 0 [ ] ir m+ 2π = (m + )i ei(m+)t = 0 (m ) 0 2π i dt = 2πi (m = ). (i) z(t) = α + re it = z (t) = ire it. (ii) m + = p 0 b a e ipt dt = 0 [ ] b ip eipt = ( e ipb e ipa). a ip (i) z(t) = (α + r cos t) + ir sin t z (t) = r sin t + ir cos t = ir(cos t + i sin t) = ire it. (ii) b a e ipt dt = b a (cos pt + i sin pt) dt = [ p ] b [ ] b (sin pt i cos pt) = a ip eipt. a

50 , (piecewise smooth), 2,..., N = + + N 8-3. f, g () α, β (αf(z) + βg(z)) dz = α f(z) dz + β g(z) dz. (2) = + 2 (3) f(z) dz = f(z) dz = f(z) dz + f(z) dz. 2 f(z) dz. (4) l() f(z) M f(z) dz M l(). 2

51 () (3) (4) = {z k } {ζ k } Σ(f, ) = f(ζ k )(z k z k ) f(ζ k ) z k z k M z k z k Ml() k k k δ( ) 0 (4) 8.2. z m vs. z m. z = z = i z m dz z m dz m 8- : z = z(t) = e (π t)i (t [0, π]) 2 : z = z(t) = t (t [, ]) 3 : z = z(t) = e ti (t [ π, 0]) 4 : z = z(t) = t + ( t )i (t [, ]) 5 : z = z(t) = t + ( t )i (t [, ]) t := z m dz, := z m dz (k =, 2, 3, 4, 5) k k k k z m = = 2 3 = 4 = 5 = 2 m+ (m ) 0 (m ).

52 () P (z) = a m z m + + a z + a 0 P (z) dz = P (z) dz = = P (z) dz 2 5 g(z) = z m = 3 2, 4, 5 (auchy s Integral Theorem): D D f : D f(z) dz = 0.. (closed curve) : z = z(t) (a t b) z(a) = z(b) (simple closed curve, s.c.c.) : z = z(t) (a t b) a < t < t 2 < b = z(t ) z(t 2 ) D (domain) D 2 D Jordan 3 (interior) 2 3 +( ) = (0,) = 0 3 = = 3 T T

53 (exterior) D (simply connected) D D D. P (z) D = 2 = 2 + ( ) = = 0 = 2 +( ) 2 = 8-4 ( : D f : D α, β D α β, 2 D f(z) dz = 2 f(z) dz i + ( 2 ) A = 0 = =. +( 2 ) 2 2 4

54 D 2

55 9 9. m Z, = (α, r) r > 0 0 (m ) (z α) m dz = 2πi (m = ) (auchy s Integral Theorem): D D f : D f(z) dz = 0.. D D = { < z < 3} f(z) = /z = (0, 2) f(z) dz = 2πi 0 : D f : D α, β D α β, 2 D f(z) dz = f(z) dz. 2. α D z D α z D z F (z) := f(ζ) dζ z z F (z) = z α f(ζ) dζ F (z) F (α) = 0 F (z) = f(z) D 49

56 f(z) G (z) = f(z) G f (primitive function) F (z) G(z) = F (z) + ( ) z α f(ζ) dζ = F (z) F (α) = G(z) G(α) z 2 z 3 /3 0 + i z 2 dz = +i 0 z 2 dz = [ z 3 3 ] +i 0 = ( + i) z α dz = 2πi α 0 α α 2. α D = 0. α r > 0 r := (α, r) dz = 2πi. r z α 0 = + 2 = 2 α, 2 = = = r r = 2πi.

57 ,... n = n = (i, ) z dz = π 2 ( ) z = 2 2i z 2i z + 2i ( ) z dz = 2 2i z 2i dz z + 2i dz. 2i 2i 2 (2πi 0) = 2i π 2 2. a x dx = π 2 a x = 2 tan θ R > R 2 I R = R x dx R := { Re it t [0, π] } R z dz + I R = π 2 R > 2 z R z z 2 2 = R 2 2 > 0 (4) z dz R 2 2 l( R) = πr R 2 0 (R ) 2 R I R π/ 2. α, β α β α + β α + β α = z 2, β = 2

58 R 2 3 R 2. f(x + yi) = u(x, y) + v(x, y)i f(z) dz = ( ) ( ) u dx v dy + i v dx + u dy.. = {z k } k f(ζ k)(z k z k ) ζ k z k z k f(ζ k ) = u k + v k i z k = x k + y k i f(ζ k )(z k z k ) = (u k + v k i){(x k x k ) + i(y k y k )} k k x k = x k x k, y k = y k y k (u k x k v k y k ) + i (v k x k + u k y k ) k k max z k 0 max { x k, y k } 0 (u k x k v k y k ) u dx v dy k b. f(z)dz = f(z(t)) z (t) dt b a u(x(t), y(t)) x (t) dt. a u(x, y) dx = (Green s Theorem) R 2 Ω P (x, y) Q(x, y) Ω P dx + Q dy = ( P y + Q x ) dx dy Ω

59 = y = ϕ(x) P (x, y) P dx = P dx + P dx + 2 P dx 3 = b = = a b P (x, c) dx + a b a = a b P (x, ϕ(x)) dx + {P (x, ϕ(x)) P (x, c)} dx + 0 { } ϕ(x) P y (x, y) dy dx Ω c P y dx dy a a P (a, y) dx x Q dy = Q x dx dy R 2 D f f(x + yi) = u(x, y) + iv(x, y) u, v D u x = v y, v x = u y D Ω (P, Q) = (u, v) (v, u) u dx v dy = ( u y v x ) dx dy Ω v dx + u dy = ( v y + u x ) dx dy Ω Ω

60 f(z) dz 0

61 0 0. (auchy s Integral Formula): D D f : D α ( ) f(α) = 2πi f(z) z α dz α f(z)/(z α) 2πif(α) α f(α). α r > 0 = (α, r) f(z) z α dz = f(α) z α dz + f(z) f(α) z α dz = 2πif(α) + f(z) f(α) z α dz. 55

62 M(r) = max f(z) f(α) z f r 0 M(r) 0 z z α = r f(z) f(α) z α M(r) 8-3(4) r f(z) f(α) z α dz M(r) r l() = 2πM(r) 0 (r 0) r 0. = (0, 2) z 2 () I = z i dz (2) e z I = z 2 4z + 3 dz () D = f(z) = z 2, α = i D f D i ( ) f(i) = 2πi z 2 dz I = 2πif(i) = 2πi. z i e z (2) z = 3 0 (z )(z 3) f(z) = ez z 3, α = D 2.5 f f() = e z 2πi (z )(z 3) dz e I = 2πif() = 2πi 3 = πei f D α ( ) f (n) (α) = n! 2πi f(z) dz. (z α) n+

63 α D n = 0 n = n 2 z 0 n = 0 f(α + z) f(α) = { } z z f(z) 2πi z (α + z) dz f(z) z α dz = f(z) 2πi (z α z)(z α) dz z 0 ( ) f (α) = 2πi. f(z) (z α) 2 dz. ( ) g n (z) (n N) g(z) g n (z) dz g(z) dz g n (z) g(z) max z g n(z) g(z) 0 (n ). = (0, 2) e z I = (z ) dz 2 () D = f(z) = e z, α =, n = 0- ( ) f () = 2πi e z (z ) 2 dz I = 2πif () = 2πie. 0.3

64 Liouville s Theorem The Fundamental Theorem in Algebra a n z n + a n z n + + a z + a 0 = 0 (a n 0, n ) 2 f : M f(z) M ( z ) f e z sin z, cos z sin x ( x R) α, α 2,... α n a n z n + a n z n + + a z + a 0 = a n (z α )(z α 2 ) (z α n ) x 2 + = 0. f : M f(z) M ( z ) α = (α, r) (r > 0) 0- f (α) = f(z) 2πi (z α) 2 dz 2π M r 2 l() = M r. M r > 0 f (α) = 0 α f. f(z) f(z) 0 z g(z) := f(z) z 0 f(z) = z n a n + a n z + + a 0 z n 2

65 A(z) z r > 0 A(z) a n + + a 0 a n z z n r + + a 0 r n r A(z) a n /2 r z r f(z) z n ( a n a n 2 ) rn a n 2 g(z) 2 r n =: M. a n E = {z z r} g(z) E M g(z) max{m, M } 0.4 (Maximum Modulus Principle) D f(z) f(z) D α D ( ) z D {α}, f(z) < f(α). D D = D = { z } f(z) = z 2 f(z) z z = ±i f(z) f(z) D D R = [, ] f(x) = x 2 x = 0 f(x) 4 5 f(z) D

66 f(z) ( ) α D r > 0 = (α, r) D z f(z) z α < f(α) r f(α) = f(z) 2πi z α dz < f(α) l() = f(α) 2π r

67 .. 0 {z n } n= α (converges to α) z n α 0 (n ) z n α (n ) α {z n } n= (limit). {z n } n=0 S n = z 0 + z + + z n {S n } n=0 lim n S n n=0 z n n 0 z n z 0 + z + z 2 + {z n } n=0 (series) z 0 + z + z 2 + (diverge). 6

68 z n (absolute convergence) n=0 z n n=0. { z z n } n { z z n } n 0. z n = (i/2) n z 0 + z + z 2 + = + /2 + /2 2 + = 2 z 0 + z + z β < β = + β + β2 +.2 β > β = 2 = 2 =

69 D f : D α D R > 0 = (α, R) D = z ( ) f(z) = f(α) + f (α)(z α) + f (α) (z α) 2 + 2! f(z) α (Taylor expansion about α). g(z) = a n (z α) n 2 n=0. z 0 ( ) z z 0 α z α < -2 f(z) f(z) = z z 0 (z α) (z 0 α) = f(z) z α z 0 α = f(z) ( ) n z α z0 α. z α n=0 z α f(z 0 ) = 2πi f(z) dz = z z 0 { 2πi n=0 = = n=0 n=0 ( (z 0 α) n 2πi } f(z) (z α) (z n+ 0 α) n dz f (n) (α) (z 0 α) n. n! ) f(z) dz (z α) n+ 2

70 = 3. z e z = + z! + z2 2! z3 + sin z = z 3! + z5 z2 cos z = 5! 2! + z4 4! α = 0 z < R R = (0, R) -3. e z = + z + z 2 /2! + z = iθ (θ R) e iθ = (iθ) n /n! = cos θ + i sin θ e iθ. e z e 2 = n! (z 2)n n=0 e z = e 2 e z 2 = e 2 n=0. (z 2) n n! f(z) A -3 f(z) = a n (z α) n B f(z) = b n (z α) n n a n = b n. f(z) = z 2i () z = 0 (2) z = i. () -2 f(z) = 2i z 2i. 3 {g n } g n (z) gn (z) dz = g n (z) dz

71 z < z < 2 2i f(z) = 2i ( z n = 2i) (2i) n+ zn ( z < 2) n=0 (2) () f(z) = (z i) i n=0 = i z i i < z i < z i i. f(z) = i ( ) z i n = i i n+ (z i)n ( z i < ) n=0 n=0. a n (z α) n (radius of convergence) z α < R z α > R R () 2 (2).3 e z z 2 = z 2 + z + 2! + z 3! + e/z = + z + 2!z + 3!z 3 + z = 0 z 0 z = 0-4. α D = {z R < z α < R 2 } f : D n Z = (α, R), (R < R < R 2 ) a n := 2πi f(z) dz (z α) n+

72 = z D f(z) = a n (z α) n n= = + a 2 (z α) 2 + a z α + a 0 + a (z α) + a 2 (z α) 2 + f(z) α (Laurent expansion about α) = + n= n<0 n 0

73 α D = {z R < z α < R 2 } f : D n Z = (α, R), (R < R < R 2 ) a n := 2πi f(z) dz (z α) n+ z D f(z) = + a 2 (z α) 2 + a z α + a 0 + a (z α) + a 2 (z α) f(z) α (Laurent expansion about α) f D = {z 0 < z α < R 2 } k a k 0 f(z) = a k (z α) k + + a z α + a 0 + a (z α) + a 2 (z α) 2 + f z = α k (pole of order k) a k 0 k α (essential singularity) (singularity) 67

74 {a n } n Z z 0 D ϵ > 0 0 = (z 0, ϵ) D r, r 2 > 0 R < r < r 2 < R 2 0, = (α, r ), 2 = (α, r 2 ) 0 f(z) z z 0 dz = 2 f(z 0 ) = 2πi 0 2 = a m (z 0 α) m, a m = f(z) dz 2πi 2πi m (z α) m+ = a n (z 0 α) n, a n = 2πi 2 2πi n 0 f(z) dz (z α) n+ 2 z z α < z 0 α = 2πi 2πi f(z) z 0 α z α dz = 2πi z 0 α f(z) z 0 α = 2πi n 0 n 0 ( f(z) (z α) n ( ) z α n dz z 0 α = = 2πi ( n 0 ) f(z) dz (z α) n (z 0 α) n+ = n+=m ( 2πi ) (z 0 α) n+ dz ) f(z) dz (z (z α) m+ 0 α) m. 2 ez D = {0 < z < } z2 e z z = ) ( + z + z2 2 z 2 2! + z3 3! = z + 2 z + 2! + z 3! + z = 0 2

75 D = {0 < z < 2} z 2 (z 2) z 2 (z 2) = z 2 2( z/2) = ) ( 2z z2 z = 2z 2 4z 8 z 6 z = D α D f : D {α} f(z) = n Z a n(z α) n α D f(z) dz = 2πi a. α., = (α, r) r D n (z α) n = 0 n = (z α) n = 2πi f(z) dz = ( ) a n (z α) n dz = ) (a n (z α) n dz = 2πi a. n Z n Z = f(z) a a (z α) f α (residue) Res(f(z), α) 2-2 f(z) dz = 2πi Res(f(z), α) e z. z = 0 z 2 ez z = 2 z + ( ) e z + Res 2 z z, 0 =. = (0, ) 2 e z dz = 2πi = 2πi. z2

76 = (0, ) dz D = { z < 2} z 2 (z 2) ( ) D {0} dz = 2πi Res z 2 (z 2) z 2 (z 2), 0 z 2 (z 2) = ( ) + Res 4z z 2 (z 2), 0 =. 2πi 4 4 = πi 2. g(z) = z 2 z 2 (z 2) dz = g(z) dz = 2πi g (0) = 2πi z 2 4 = πi 2.. f(z) = g(z) g(z) (z α) k z = α = (α, r) r Res(f(z), α) = f(z) dz = 2πi 2πi g(z) (z α) dz = k (k )! g(k ) (α). (k ) 2-3 f(z) = Res(f(z), α) = k = Res(f(z), α) = g(α). g(z) g(z) z = α (z α) k (k )! g(k ) (α). D f D {α, α 2,, α n } {α, α 2,, α n } D n f(z) dz = 2πi Res(f(z), α k ). k=

77 r > 0 k = (α k, r),..., n D = = (0, 2) I = n e iz dz z 2 + f(z) = eiz {±i} z = ±i z 2 + f(z) dz = 2πi{Res(f(z), i) + Res(f(z), i)}. g(z) = eiz g z = i 2-3 Res(f(z), i) = z + i g(i) = e 2i. h(z) = eiz 2-3 Res(f(z), i) = h( i) = z i e 2i. ( ) e I = 2πi = π(e e). 2i e 2i. eiz z 2 + = { e iz 2i z i } eiz z + i f(z) = e /z z 2 sin z

78 D α D f : D {α} f(z) = n Z a n(z α) n α D f(z) dz = 2πi a. f(z) z = α (z α) f α (residue) Res(f(z), α) D f D {α, α 2,, α n } {α, α 2,, α n } D n f(z) dz = 2πi Res(f(z), α k ). k= I = f(z) dz (i) f(z) α,, α n (ii) Res(f(z), α k ) (iii) I = 2πi ( ) 72

79 I = 2π cos θ dθ = π 2 z = e iθ (0 θ 2π) z sin θ = z z 2i, cos θ = z + z, dz = ie iθ dθ dθ = dz 2 i z, I = =(0,) z + z 2 dz i z = 2 i 3z 2 + 0z + 3 dz f(z) f(z) = z = /3 3(z + /3)(z + 3) I = 2 ( i 2πi Res f(z), ) f(z) = Res(f(z), α) = k = Res(f(z), α) = g(α). g(z) g(z) z = α (z α) k (k )! g(k ) (α). f z = /3 k = ( Res f(z), ) = 3 3(z + 3) z= 3 = 8. I = 2 i 2πi 8 = π 2. t = tan θ/2

80 I = + x 4 dx = π 2 f(z) = + z 4 = dz α = + z4 e πi/4, β = e 3πi/4 { ( ) ( )} = 2πi Res + z, α + Res 4 + z, β f(z) = z = α g(z) Res(f(z), α) = g (z). ( ) Res + z, α = = e 3πi/4 = i 4 4α ( ) Res + z, β = = e 9πi/4 = i 4 4β ( i = 2πi i ) 4 = π. 2 2 I J R := {z = x R x R} R := { z = Re iθ 0 θ π } = J R + R. 2 Mathematica { log ( x 2 2x + ) + log ( x 2 + 2x + ) 2 tan ( 2x ) + 2 tan ( 2x + )} /(4 2)

81 lim R = J R + = π J R 2 z = R > R f(z) = π R 0 I = R 2 R z 4 + z 4 = R 4 > 0 R 4 f(z) dz l( R) R R 4 = πr R 4 0 (R ). I = π α h(z) h(α) 0 f(z) = g(z) = 2-3 Res(f(z), α) = (z α)h(z) h(α). g (z) = h(z) + (z α)h (z) g (α) = h(α) I = 2 f(z) = cos x + x 2 dx = π e eiz + z 2 z = i = 2πi Res(f(z), i). f(z) = z i e iz z + i eiz z + i Res(f(z), i) = ei i 3 = f(z) dz z = i 2-3 i + i = e 2i.

82 = 2πi e 2i = π e. 2 R 0 z = R > R z 2 + z 2 = R 2 > 0 z = x + yi R y 0 e iz = e i(x+yi) = e y R f(z) = e iz + z 2 R 2 f(z) dz l( R) R R 2 = πr 0 (R 0). R 2 R e ix R = I R R + x dx = cos x 2 R + x dx + i 2 0 I = lim R R R I R = sin x sin x dx + x2 + x 2 = π e 3.5. α () m Z, r > 0, = (α, r) (z α) m dz (2) α α) m dz (z α) m dz = (z 2. z = z = : z = z(t) = e (π t)i (t [0, π]) 2 : z = z(t) = t (t [, ]) 3 : z = z(t) = t + ( t )i (t [, ])

83 t i (i =, 2, 3) z 3 dz 3. () = (0, 2) e z z + dz (2) dz (3) z4 (z ) 2 (z + 3) e iz (3z π) 2 dz 4. ( () (2) (3) z 2 exp ) z 3 z 2 (z 2) z 5. I = 2π cos θ dθ = π 2

84 4. 4. f : D D f D f(z) dz = 0. (Morera) f : D D ( ) D f(z) dz = 0 f D. ( ) F : D F (z) = f(z) (z D) F D f f F ( ) α D ζ D α ζ D, 2 ( ) 2 + ( ) f(z) dz = 0 2 +( ) = 2 = 78

85 D α ζ F (ζ) := f(z) dz F (ζ) = ζ α ( ) = f(z) dz 2 f(z) dz F : D f ζ D F (ζ + h) F (ζ) lim = f(ζ) h 0 h ϵ > 0 δ > 0 h δ F (ζ + h) F (ζ) f(ζ) h ϵ f ϵ > 0 δ > 0 z ζ δ f(z) f(ζ) ϵ h δ F (ζ + h) F (ζ) h = h { ζ+h α ζ α } D α ζ ζ ζ + h L : z(t) = ζ + ht (0 t ) F (ζ + h) F (ζ) h = h { +L } = f(z) dz h L

86 L dz = F (ζ + h) F (ζ) h 0 h dt = h l(l) = h L f(ζ) = h L = h L dz L (f(z) f(ζ)) dz h e l(l) = ϵ f(z) dz f(ζ) h ζ D ( ). f : D F : D F = f F f (primitive function) F F 2 f F (z) F 2 (z) D f : D ( ) F : D. f(z) = z 2 F (z) = z 3 /3 + (z 3 /3) = z 2. f(z) = F (z) = log z + z log z = (0, ) dz = 2πi 0 z ( ) f(z) D = {0} D f D := (, 0]

87 ( ) F (ζ) = ζ f(z) dz D F log z Im log z < π D D(α, r) := {z z α < r} α r D(α, r) (α, r) g n g g n g g n (z) dz g(z) dz g n g g n g g n : D (n N) D E g n E g : E (uniform convergence) ( ϵ > 0)( N N)( n N)( z E) g n (z) g(z) < ϵ E g n g z 0 g n g (n ) g = lim n g n 2 Log z π Im log z < π F (ζ) = Log ζ

88 < r < g n (z) = + z + + z n, g(z) = z g n g E(r) = {z z r} z < z = + z + + zn +. g n (z) = ( z n+ )/( z) g n (z) g(z) = z n+ z rn+ r 0 (n ) r n+ /( r) z E(r) g n g E(r) = {z z r} r z 4- g n : D (n 0) g : D D g D. ϵ > 0 α D δ > 0 z α < δ = g(z) g(α) < ϵ N ( n N)( z D) g n (z) g(z) < ϵ/3 g n (α) g(α) < ϵ/3

89 m N g m D z = α δ > 0 z α < δ = g m (z) g m (α) < ϵ/3 z α < δ g(z) g(α) g(z) g m (z) + g m (z) g m (α) + g m (α) g(α) < ϵ/3 + ϵ/3 + ϵ/3 = ϵ 4-2 g n : D (n 0) g : D D D g n (z) dz g(z) dz (n ).. n N g n (z) dz ( ϵ > 0)( N N)( n N)( z ) g n (z) g(z) < ϵ. g(z) dz = (g n (z) g(z)) dz ϵ l() l() lim n g(z) dz g n (z) dz = 4-3 g n : D (n 0) g : D D () g D (2) D g n g () D D D g n D g n (z) dz = 0

90 g(z) dz = 0 4- g 3 g D (2) E D E D 4 r > 0 α E D(α, r) (α, r) D g n ( ϵ > 0)( N N)( n N)( z D) g n (z) g(z) < ϵ. 2 = (α, r) g n(α) = g n (z) 2πi (z α) 2 dz g (α) = 2πi g n(α) g (α) 2πi g(z) (z α) 2 dz g n (z) g(z) (z α) 2 dz ϵ 2π r 2 l() = ϵ r. ϵ/r α E g n g E. g n : D g : D E D g n E g E 4-3 g n g D g g n g D (Weiserstrass s Theorem) g n (z) = + z + z z n g(z) = /( z) D(0, ) g (z) = lim n g n(z) ( z) 2 = + 2z + + nzn + sin nx g n (x) = (x R) n g n g(x) = 0 R g n(x) = cos nx 3 g n 4 {e n } E e n D /n { e n(k) } z D E z E E D D D D =

91 g (x) = 0 z g n (z) g(z)

92 5 5. f n : D (n = 0,, 2,...) g n (z) := f 0 (z) + f (z) + + f n (z) z D g n (z) g(z) g(z) = f 0 (z) + + f n (z) +, g(z) = f n (z), g(z) = f n (z) n=0 n 0 g : D 5- () D g n (z) = f 0 (z) + f (z) + + f n (z) g(z) = f n (z) n=0 f n (z) dz = f n (z) dz. n=0 n=0 86

93 (2) g n (z) = f 0 (z) + f (z) + + f n (z) D g(z) = f n (z) n=0 ( f n (z)) = n=0 () g = lim n g n = g(z) dz = lim g n (z) dz = lim n n f n(z). n=0 n f n (z) dz = lim k=0 = 4-2 n n k=0 f n (z) dz = (2) D f : D α D R > 0 = (α, R) D z f(z) = f(α) + f (α)(z α) + f (α) (z α) 2 + 2! D z 0 r := z 0 α < R z z 0 α z α = r R < f(z) z z 0 = f(z) (z α) (z 0 α) = f(z) z α z 0 α z α = f(z) z α n=0 ( ) z0 α n. z α f n (z) := f(z) z α ( ) z0 α n g n (z) := f 0 (z) + + f n (z) g(z) := f(z) z α z z 0 z g(z) = lim g n (z) = f n (z) n=0

94 f(z) M = M() z β = z 0 α z α g n (z) g(z) = f n+ (z) + f n+2 (z) + = f(z) z α (βn+ + β n+2 + ) M R β n+ β M β n+ R( β ) β = r/r < z g n g f(z 0 ) = 2πi f(z) dz = z z 0 2πi = 2πi n=0 = 2πi = = n=0 n=0 n=0 f n (z) dz n=0 ( 2πi f n (z) dz 5-() ( ) f(z) z α z0 α n dz z α ) f(z) dz (z (z α) n+ 0 α) n f (n) (α) (z 0 α) n. n! 0 < r < R E r := {z z α r} r < r < R z 0 α = r z 0 f(z 0 ) = a n (z 0 α) n a n = f (n) (α)/n! 5-3 n=0 α f(z) = 5-3. a n (z α) n E(r) n=0 F (z) = a 0 + a z + + a n z n + z 0 z < z 0 z F (z) 0 < r < z 0 F n (z) = a 0 + a z + + a n z n E(r) := {z z r}. r < z 0 r F (z 0 ) = a 0 + a z a n z n 0 + a n z0 n 0 (n ) M > 0 n 0 a n z0 n M z r < z 0 z a n z n = a n z0 n z n ( ) r n z 0 M z 0 { lim ( a 0 + a z a n z n n 0 ) lim M + r ( ) r n } n z M z 0 r/ z 0.

95 F (z) z r z F (z) r (< z 0 ) z < z 0 F (z) E(r) z E(r) F (z) F n (z) = a n+ z n+ + a n+2 z n+2 + = lim a n+z n+ + + a n+n z n+n N { ( ) r n+ ( ) } r n+n lim M + + M (r/ z 0 ) n+ N z 0 z 0 r/ z 0. z n E(r) F α D = {z R < z α < R 2 } f : D n Z = (α, R), (R < R < R 2 ) a n := 2πi f(z) dz (z α) n+ z D f(z) = + a 2 (z α) 2 + a z α + a 0 + a (z α) + a 2 (z α) D z r 2 < R 2 z r > R F (z) = a 0 + a /z + + a n /z n () 5.3. u(x, y) dx 2

96 ( ) u(x, y) V = V (x, y) := R 2 v(x, y) V : R 2 R 2 field (field) { ( ) ( ) } x x(t) = p = = t [a, b] y y(t) = {p k } k p k := p k p k 2 Σ(V, ) = k V (p k ) p k ( ) ( ) uk xk V (p k ) = p k = v k y k Σ(V, ) = ( uk k v k ) ( ) xk = (u k x k + v k y k ) y k k V (x, y) = (cos xy, sin(x + y)) δ( ) := max p k 0 k V V dp = u(x, y) dx + v(x, y) dy u(x, y) dx v(x, y) dy v(x, y) 0 u(x, y)dx + 0 dy. f = f(p) = f(x, y) R 2 f R (scalar field) ( ) x f p = y f(x + x, y + y) = f(x, y) + a x + b y + o( x 2 + y 2 ) ( ) a ( a = f x (x, y), b = f y (x, y) ) V = b f(p + p) = f(p) + V p + o( p ) V p p

97 f p V p V V p f f p (gradient vector) V = grad f(p) grad f(p) := (f x (x, y), f y (x, y)). f grad f grad f h p grad h grad h. ( ) u(x, y) V = V (p) = v(x, y) rot V (p) := u y (x, y) + v x (x, y) R V (rotation) V p 2ϵ v(x + ϵ, y) u(x, y + ϵ) v(x ϵ, y) + u(x, y ϵ) 2 rot V (p) ϵ + o(ϵ) u, v u(x, y+ϵ) = u(x, y)+u y (x, y)ϵ+o(ϵ) 2{ u y (x, y) + v x (x, y)}ϵ + o(ϵ) = 2 rot V (p) ϵ + o(ϵ). Ω V V dp = rotv dx dy V (P (x, y), Q(x, y)) D. V div V (p) := u x (x, y) + v y (x, y) R V (divergence) p ϵ > 0 div V (p) ϵ 2 + o(ϵ 2 ) 3 3

98 = {p(t) = (x(t), y(t)) t [a, b]} Ω ( ) ( ) u(x, y) v(x, y) V = V = v(x, y) u(x, y) div V = rot V V dp = div V dx dy. p(t) p (t) = (y (t), x (t)) V dp = b a D V (p(t)) p (t) dt = b a V (p(t)) p (t) dt

99

100 z = 2 i, w = 3 + 2i () z + w (2) z w (3) zw (4) z/w () () ( + i) 5 (2) 2i(2 + i)(2 + 4i)( + i) (3) -3. (3 + 4i)( i) 2 i -2() z, w zw = 0 z = 0 w = a, b, c, d az 3 + bz 2 + cz + d = 0 z = α α z = + 3 i z, z 2, z 3, z 4 z = r(cos θ + i sin θ) 0 () z = r{cos( θ) + i sin( θ)} (2) z = {cos( θ) + i sin( θ)} r (2) -2(2) -. de Moivre -3-2 Hint: n 0 n < 0-3 (de Moivre ) z = r(cos θ + i sin θ) n z n = r n (cos nθ + i sin nθ) -2. z z > /z z 2 z w w = /z 0 < z < z =

101 () + i (2) 5 3i (3) 4 + 7i (4) 8 i 3-2. () ( + i) 5 = + i 5 = ( 2) 5 = 4 2. (2) 2i(2 + i)(2 + 4i)( + i) = = (3) (3 + 4i)( i) 3 + 4i i 2 i = = 5 2 = 0. 2 i zw = 0 zw = 0 z = 0 w = 0 z = 0 w = 0. z = a + bi, w = c + di, () (z) = a bi = a + bi = z. (2). (3) z w = (ac bd) + (ad + bc)i = (ac bd) (ad + bc)i = (a bi)(c di) = z w. (4). (5) ( ) = (6)(7). (a + bi) + (a bi) 2 = a = Re z. -5., z = α aα 3 + bα 2 + cα + d = 0., a, b, c, d α. aα 3 + bα 2 + cα + d = 0 aα 3 + bα 2 + cα + d = ( z = 2 cos π 3 + i sin π ) (, z 2 = 4 cos 2π i sin 2π ), ( 3 z 3 = 8 (cos π + i sin π), z 4 = 6 cos 4π 3 + i sin 4π ) () cos( θ) = cos θ, sin( θ) = sin θ z = r(cos θ i sin θ) = r{cos( θ) + i sin( θ)}. (2) z = cos θ i sin θ r(cos θ + i sin θ) z = cos θ i sin θ r(cos 2 θ + sin 2 θ) = r (cos θ i sin θ) = {cos( θ) + i sin( θ)}. r

102 , α, β α 2 αβ + β 2 = 0 (Hint. Make use of the conditions α = β and arg α arg β = ±π/3: then you ll find the polar representation of α/β.) 2-2. z + : z 2 = 3 : z (Hint. z a 2 = (z a)(z a) = (z a)(z a) ) 2-3. () e 2+ π 4 i (2) e 3+πi (3) e log 3 3π 2 i () z (e z ) = e z. (2) z w e z+w = (e z ) (e w ) 2-5. N () z 4 = 6i (2) z 4 + z 3 + z 2 + z + = 0 2- N. z N = w w N 2-3 w 0

103 α, β 0 : 0, α, β 0 α = β arg α arg β = ± π 3 α β = arg α β = ± π 3 α β = cos(± π 3 ) + i sin(± π 3 ) = ± 3i 2. α 2 αβ + β 2 = 0 ( α β )2 ( α β ) + = 0 α β = ± 3i z + : z 2 = 3 : 3 z 2 = z (z 2)(z 2) = (z + )(z + ) 8zz 9z 9z + 35 = 0. ( z 9 ) ( ) ( 8 z 9 8 = 9 ) 2 8 z 9 8 = 9 8. z 9 8, ( () e 2+ π 4 i = e 2 cos π 4 + i sin π ) ( = e i ). 4 2 (2) e 3+πi = e 3 (cos π + i sin π) = /e 3. (3) e log 3 3π 2 i = 3 { cos ( 3π 2 ) + i sin ( )} 3π = 3i () z = x + yi (x, y R), e z = e x (cos y + i sin y) = e x (cos y i sin y) = e x {cos( y) + i sin( y)} = e x yi = e z. (2) -(2) e z+w = e z+w = e z e w = e z e w () z = re θi (r > 0, 0 θ < 2π), z 4 = r 4 e 4θi. 6i = 6e π/2+2nπi (n Z) r = 2, θ = π 8 + π n (n = 0,, 2, 3). 2 z = 2e π 8 i, 2e 5π 8 i, 2e 9π 8 i, 2e 3π 8 i. (2) z 5 = (z )(z 4 + z 3 + z 2 + z + ). z 5 = z = 2.6 z = e 2mπi 5 (m =, 2, 3, 4) e 2πi 5, e 4πi 5, e 6πi 5, e 8πi 5.

104 ( ). w = f(z) = e z 2 () f(s ), S = {x + yi 0 x, π/2 y π/2} (2) f(s 2 ), S 2 = {x + yi log 2 x log 2, 5π y 5π} (3) f (T ), T = {u + vi {0} 0 < u, v = 0}, 3-2( ). () log( + 3i) (2) log( 2) (3) log i (4) log e 3-3. () ( + 3i) i (2) ( 2) +i (3) i /3 3-. z 0 m z m z m z, w log zw = log z + log w Log zw = Log z + Log w 2 f : S, T f(s) := {f(z) : z S} f S f (T ) := {z : f(z) T }. f T

105 () z = log( + 3 i) e z = + 3 i = 2e πi/3 = e log 2 e (/3+2m)πi (m Z) z = log 2 + (/3 + 2m)πi (m Z). (2) z = log( 2) e z = 2 = 2e πi = e log 2 e (+2m)πi (m Z) z = log 2 + ( + 2m)πi (m Z). (3) z = log i e z = i = e πi/2 = e (/2+2m)πi (m Z) z = (/2 + 2m)πi (m Z). (4) z = log e e z = e = e e 2mπi (m Z) z = + 2mπi (m Z) () ( + 3 i) i = e i log(+ 3 i) = e i{log 2+(/3+2m)πi} (m Z) = e (/3+2m)π+i log 2 (m Z) = e (/3+2m)π {cos(log 2) + i sin(log 2)} (m Z). (2) ( 2) +i = e (+i) log( 2) = e (+i){log 2+(+2m)πi} (m Z)

106 = e {log 2 (+2m)π}+{log 2+(+2m)π}i (m Z) = 2 e (+2m)π e i log 2 e πi (m Z). = 2 e (2k+)π {cos(log 2) + i sin(log 2)} (k Z). (3) i /3 = e (/3) log i = e (/3) (/2+2m)πi (m Z) = e (/6+2m/3)πi (m Z) = e πi/6, e 5πi/6, e 3πi/2 = 3 + i 2, 3 + i, i. 2

107 z = x + yi w = u + vi z + w z + w z w z + w (Hint. Apply the triangle inequality for z = z and w = z + w.) () sin ( π 4 + i) (2) sin(x + yi) x, y R x, y 4-4. () cos z = 0 z = ( 2 + m) π (m Z) (2) cos z 0 z tan z := sin z cos z (a) tan (z + π) = tan z (b) i tan i = e2 + e () cos ( π 2 z) = sin z (2) sin ( π 2 z) = cos z (3) sin z = sin z (4) cos z = cos z (5) e iz = cos z + i sin z 4-2. M > 0 z sin z M cos z M (Hint: 2 cos z = e iz + e iz e iz e iz ) 4-3. g(z) = z N N f(z) = z N z = 0

108 z + w z + w z + w 2 z z w + w 2 2 (x + u) 2 + (y + v) 2 x 2 + y z w + u 2 + v 2 xu + yv z w. xu + yv < 0 xu + yv 0 xu + yv z w (xu + yv) 2 (x 2 + y 2 )(u 2 + v 2 ) 0 (xv yu) 2 xu + yv 0 xv = yu z = 0 w = 0 zw 0 k > 0 z = kw z = z, w = z +w z +(z +w ) z + z +w w z w + z. z, w z w z + w. 2. Re z z z z + w 2 = (z + w)(z + w) = zz + zw + wz + ww = zz + 2Re (zw) + ww zz + 2 zw + ww = z z w + w 2 = ( z + w ) 2. Re (zw) = zw zw 4-2. () cos(z + 2π) = e(z+2π)i + e (z+2π)i = ezi + e zi = cos z. (sin z ) ( 2 2 e (2) cos 2 z +sin 2 zi + e zi ) 2 e z = +( zi e zi ) 2 = e2zi e 2zi e2zi 2 + e 2zi =. 2 2i 4 4 (3) cos z cos w sin z sin w = ezi + e zi ewi + e wi ezi e zi ewi e wi 2 2 2i 2i = e(z+w)i + e (z w)i + e (z w)i + e (z+w)i + e(z+w)i e (z w)i e (z w)i + e (z+w)i 4 4 = e(z+w)i + e (z+w)i = cos(z + w). 2 sin(z + w)

109 () sin( π 4 + i) = e( π 4 +i)i e ( π 4 +i)i = ( {e cos π 2i 2i 4 + i sin π ) ( e cos π 4 4 i sin π )} 4 = { ( e 2 + ) 2 i e( 2i )} i = ( + e2 ) ( e 2 )i 2 2 2e. 2 (2) sin(x + yi) = sin x cos(yi) + cos x sin(yi) = e y + e y sin x + i ey e y cos x. 2 2 Re sin(x + yi) = e y + e y sin x, Im sin(x + yi) = ey e y cos x () ezi + e zi = 0 e zi = e zi e 2zi = = e πi+2mπi (m Z). 2 2zi = πi + 2mπi z = ( + m)π (m Z). 2 sin(z + π) sin z cos π + cos z sin π (2)(a) tan(z + π) = = cos(z + π) cos z cos π sin z sin π = sin z = tan z. cos z e ii e ii (b) i tan i = i 2i e ii + e ii = e2 + e 2. 2

110 f(z) f () () f(z) = iz + z 2 (2) f(z) = /z 3 (3) f(z) = z 2z 5-2. () f(z) = iz + z 0 (2) f(z) = z z 2 i (3) f(z) = (z 2 i) f(z) = Re z z 5-4. z 0 f(z) = z 2 f(z) z = e z (e z ) = e z () (sin z) = cos z (2) (cos z) = sin z 5-. w = f(z) z = α z = α

111 f(z) f() zi + z 2 (i + ) 5-. () lim = lim z z z z (z )i + (z + )(z ) = lim = lim{i + (z + )} = 2 + i. z z z f(z) f() (2) lim = lim z z z z 3 z = lim z (z 2 + z + ) z z 3 z 3 (z ) = lim z (z )(z 2 + z + ) z 3 (z ) = lim z 3 = 3. (3) z = re θi (r 0, θ : ), z = re θi +, z = re θi +, f(z) f() (z 2z) ( ) = = reθi + 2(re θi + ) + z z re θi = reθi 2re θi re θi = 2e 2θi. θ z r () i + 0z 9, (2) z2 + i (z 2 i) 2 (3) 0z(z 2 i) z α = re θi (r 0, θ : ), z = re θi +α, z = re θi +α, f(z) f(α) z α z+z 2 α+α 2 (z α) (z α) = = = reθi + re θi z α 2(z α) 2re θi = e 2θi. θ z r z α = re θi (r 0, θ : ), z = re θi + α, f(z) f(α) = z2 α 2 (z α)(z + α) = = re θi( re θi + 2α ) z α z α z α re θi = e 2θi (re θi + 2α). α = 0, e 2θi (re θi + 0) = re 3θi 0 (r 0), z = 0., α 0, θ e 2θi (re θi + 2α) 2e 2θi α (r 0) θ z 0. ( e 5-5. () (sin z) zi e zi ) = = iezi + ie zi = ezi + e zi = cos z. 2i 2i 2 ( e (2) (cos z) zi + e zi ) = = iezi ie zi = ezi e zi = sin z i

112 () e z sin z (2) ze z2 (3) tan z (4) cos(z + z 2 ) 6-2. f : D f(x + yi) = u(x, y) + v(x, y)i () D =, f(z) = z 2 (2) D = {0}, f(z) = /z 6-3. f(x + yi) = u(x, y) + v(x, y)i () f(x + yi) = (x 3 3xy 2 ) + i(3x 2 y y 3 ) (2) f(x + yi) = e y (cos x + i sin x) 6-4. f(x + yi) = (x 3 3xy 2 ) + v(x, y)i v(x, y) = 3x 2 y y 3 + ( ) F : (x, y) (u, v) = (u(x, y), v(x, y)) f(x + yi) = u(x, y) + v(x, y)i () F : (x, y) (u, v) = (x, y) (2) F : (x, y) (u, v) = (x, 2y) (3) F : (x, y) (u, v) = (x 2 + y 2, 2xy) (4) F : (x, y) (u, v) = (x, 0) 6-6. () f(z) = e z (2) f(z) = z 2 (3) f(z) = z 2 + z D f(z) f(z) () Re f(z) (2) Im f(z) (3) f(z) (4) z D f (z) = f : D z = x+yi D f(x + yi) = u(x, y) + v(x, y)i f x y f x (z) := lim x 0 f((x + x) + yi) f(z) f(x + (y + y)i) f(z), f y (z) := lim x y 0 y f y = if x

113 () (e z sin z) = e z cos z + e z sin z = e z (sin z + cos z) (2) (ze z2 ) = e z2 + ze z2 2z = e z2 ( + 2z 2 ) (3) (tan z) = ( sin z cos z ) = cos2 z+sin 2 z cos 2 z = cos 2 z (4) (cos(z + z 2 )) = sin(z + z 2 ) (z + z 2 ) = ( + 2z) sin(z + z 2 ) 6-2. () z = ( x+yi ), ( ) ( f(z) = (x+yi) ) 2 = x ( 2 y 2 +2xyi. ) x u x 2 u(x, y) = x 2 y 2 y 2 ux u y, v(x, y) = 2xy =. = y v 2xy v x v y ( ) 2x 2y. u x = v y, u y = v x. 2y 2x (2) z = x + yi, f(z) = x+yi = x yi. u(x, y) = x, v(x, y) = y x 2 +y 2 x 2 +y 2 x 2 +y ( ) ( ) ) ( ) 2 y x u ux u 2 x 2 2xy y =. = (x 2 +y 2 ) 2 (x 2 +y 2 ) 2. u x = v y, y v ( x x 2 +y 2 y x 2 +y 2 u y = v x. v x v y 2xy y 2 x 2 (x 2 +y 2 ) 2 (x 2 +y 2 ) 2 ( ) ( ) x u 6-3. () f(x + yi) = (x 3 3xy 2 ) + i(3x 2 y y 3 ) = y v ( ) ( ) ( ) x 3 3xy 2 ux u y 3x 2 3y 2 6xy. =.,. 3x 2 y y 3 6xy 3x 2 3y 2 v x v y u, v., 8-2 f(x + yi) f (z) = u x + v x i = 3(x 2 y 2 + 6xyi. f(z) = z 3, f (z) = 2z 2. ( ) ( ) ( ) ( ) x u e y (2) f(x + yi) = e y cos x ux u y (cos x + i sin x) =. = y v e y sin x v x v y ( ) e y sin x e y cos x.,. u, v. e y cos x e y sin x, 8-2 f(x + yi) f (z) = u x + v x i = e y (sin x i cos x). f(z) = e iz, f (z) = ie iz f(x+yi) = u(x, y)+v(x, y)i. f(x+yi) f ( ) ( ) ( ) ( ) ux u y 3x 2 3y 2 6xy vx 6xy. = =. v y 3x 2 3y 2 v x v y v x y v = 3x 2 y y 3 +g(x). x v x = 6xy + d dxg(x) = 6xy g(x) R. f(x + yi) = u + vi = (x 3 3xy 2 ) + (3x 2 y y 3 + )i, v(x, y) = 3x 2 y y 3 + i ( ). v y v y

114 ( ) ( ) ux u y () =. u x v y v x v y 0. ( ) ( ). ux u y 0 (2) =. u x v y. v x v y 0 2. ( ) ( ) ux u y 2x 2y (3) =. u y v x v x v y 2y 2x. (. ) ( ) ux u y 0 (4) =. u x v y. v x v y () z = x+yi ( ), ( ) f(z) ( = e x yi = ) e x (cos y i ( sin y). ) u(x, y) ( = e x cos y, v(x, y) = ) e x sin y x u e x cos y ux u y e x cos y e x sin y =. =. y v e x sin y v x v y e x sin y e x cos y, f(z). (2) z = x + yi, f(z) = x + yi 2 = x 2 + y 2. u(x, y) = x 2 + y 2, v(x, y) = 0 ( ) ( ) ( ) ( ) ( ) x u x 2 + y 2 ux u y 2x 2y =. =. y v 0 v x v y 0 0 (x, y) = (0, 0). f(z) z = 0. f(z). (3) z = x + yi, ( ) f(z) ( = ) (x + ( yi) 2 + (x ) yi) 2 = 2(x ( 2 y 2 ). ) u(x, ( y) = 2(x ) 2 y 2 ), x u 2(x 2 y 2 ) ux u y 4x 4y v(x, y) = 0 =. =. y v 0 v x v y 0 0 (x, y) = (0, 0). f(z) z = 0. f(z).

115

116 : p = p(t) = (t, 0) (t [, ]) 2 : p = p(t) = (t 2, t) (t [0, ]) 3 : p = p(t) = (cos t, sin t) (t [0, 2π]) () dx + dy (2) 8-2. x dx y dy (3) (x 2 y 2 ) dx 2xy dy : z = z(t) = ( + i)t (t [0, ]) 2 : z = z(t) = t + it 2 (t [0, ]) () z 2 dz (2) (z ) dz (3) ( z ) dz (4) Re z dz (5) 8-3. z 2 dz (α, r) = { z(t) = α + re it t [0, 2π] } t () z m dz = (0, r), r > 0 m (2) (3) z m dz = (0, r), r > 0 m { (z 8) 4 + 8(z 8) } z (z 8) 4 dz = (8, 4), r > z = z = : z = z(t) = e (π t)i (t [0, π]) 2 : z = z(t) = t (t [, ]) 3 : z = z(t) = e ti (t [ π, 0]) 4 : z = z(t) = t + ( t )i (t [, ]) 5 : z = z(t) = t + ( t )i (t [, ]) t i (i =, 2, 3, 4, 5) m z m dz z m dz Hint: z m dz m = {z(t) t [a, b]} α = z(a), β = z(b) dz = dz = β α

117 dx = dt, dy = 0 dt. : (x(t), y(t)) = (t, 0) (x (t), y (t)) = (, 0). 2 : (x(t), y(t)) = (t 2, t), (x (t), y (t)) = (2t, ). dx = 2t dt, dy = dt. 3 : (x(t), y(t)) = (cos t, sin t), (x (t), y (t)) = ( sin t, cos t). dx = sin t dt, dy = cos t dt. () dx + dy = ( dt + 0 t) = dt = 2. dx + dy = (2t dt + dt) = (2t + ) dt = π 2π dx + dy = ( sin t dt + cos t dt) = ( sin t + cos t) dt = (2) x dx y dy = (t dt + 0 dt) = t dt = 0. x dx y dy = (t 2 2t dt t dt) = (2t 3 t)dt = π 2π x dx y dy = (cos t ( sin t) dt sin t cos t dt) = sin 2t dt = (3) (x 2 y 2 )dx 2xy dy = (t 2 dt 0 dt) = t 2 dt = 2 3. (x 2 y 2 )dx 2xy dy = ((t 4 t 2 ) 2tdt 2t 3 dt) = (2t 5 4t 3 ) dt = π (x 2 y 2 )dx 2xy dy = {(cos 2 t sin 2 t) ( sin t)dt 2 cos t sin t cos t dt} 3 0 = 2π (cos 2t sin t + sin 2t cos t)dt = 2π 0 sin 3t dt = 0. : z = z(t) = ( + i)t, dz = ( + i) dt. 2 : z = z(t) = t + t 2 i, dz = ( + 2ti) dt. () z 2 dz = {( + i)t} 2 ( + i) dt = ( + i) 3 t 2 ( + i)3 dt = = i. z 2 dz = (t + t 2 i) 2 ( + 2ti) dt = (t 2 + 4t 3 i 5t 4 2t 5 i) dt = i. (2) (z ) dz = {( + i)t } ( + i) dt = (2it i ) dt =. 0 0 (z ) dz = (t + t 2 i ) ( + 2ti) dt = ( 2t 3 + 3t 2 i 2ti + t ) dt = z 2, z [ ] z z 3 /3, z 2 3 +i ( + i)3 /2 z = = =, = = [ ] 2 z 2 +i 2 z = 0

118 (3) (z ) dz = {( i)t } ( + i) dt = (2t i) dt = i. 0 0 (z ) dz = (t t 2 i ) ( + 2ti) dt = (2t 3 + t 2 i 2ti + t ) dt = i. (4) Re z dz = t ( + i) dt = + i. 0 2 Re z dz = t ( + 2ti) dt = i. (5) z 2 dz = {( i)t} 2 ( + i) dt = ( i) 2 ( + i) t 2 dt = 2 2i z 2 dz = (t t 2 i) 2 ( + 2ti) dt = (t 2 + 3t 4 2t 5 i) dt = i = (0, r) = {z(t) = re ti t [0, 2π]}, dz = rie ti dt. 2π 2π () z m dz = (re ti ) m rie ti dt = ir m+ e t(m+)i dt 0 [ ] 0 2π ir m+ e t(m+)i = 0 (m ) = (m + )i 0 2π i dt = 2πi. (m = ) 0 2π 2π (2) z m dz = (re ti ) m rie ti dt = ir m+ e t( m)i dt 0 [ ] 0 2π ir m+ e t( m)i = 0 (m ) = ( m)i 0. 2π ir 2 dt = 2πr 2 i. (m = ) 0 (3) = (8, 4) = {z(t) = 8 + 4e ti ; t [0, 2π]}. z 8 = ζ, ( ) = (ζ 4 + 8ζ (0,4) ζ + 8 ζ 4 ) dζ = 4 dζ = 8πi. () (0,4) ζ

119 () e z dz (2) (sin z + 3 cos z) dz (3) e z (4) dz (5) z 5 z 2 dz (6) () dz (2) z(z 2i) p(z) dz (p(z) z ) sin(z + 3i) dz 4z 2 dz (3) + z 2 + dz (α, r) = { z(t) = α + re it t [0, 2π] } 2z 2 + 3z 2 dz () = (i, ) (2) = ( i, ) (3) = (0, 2) (4) = (, ) 9-3. a, b E E = {z(t) = a cos t + ib sin t t [0, 2π]} 2π 2π () dz (2) z a 2 cos 2 x + b 2 sin 2 dx = x ab H(r) = { z(t) = re it t [0, π] } t () 0 < r < R : H(r) e iz z dz e iz R dz = 2i H(R) z r sin x x dx. sin x (2) 0 x = π 2

120 () e z. 0. (2) sin z + 3 cos z. () (sin z + 3 cos z) dz = 0. (3) p(z) z. () p(z) dz = 0. e z dz = (4) z = 5. z 5 dz = 0. z 5 e (5) z z 2 8 z = ±2 2. z = ±2 2. e z z 2 dz = 0. 8 (6) sin(z + 3i) = 0 z. sin w = 0 w = nπ (n Z) sin(z + 3i) z = nπ 3i. z = nπ 3i e z z 2 dz = ( () z(z 2i) dz = 2i z ) z 2i dz = i 2 { z dz } z 2i dz. z = 0, z = 2i, = i (2πi 0) = π. ( ) 2 (2) 4z 2 + dz = 4i z i 2 z + i dz. z = ±i/2, 2 = (2πi 2πi) = 0. 4i ( (3) 2z 2 + 3z 2 dz = 5 z 2, = 5 (2πi 0) = 2 5 πi. ) dz. z = z z = ( z 2 + dz = 2i z i ) dz. z + i () = (i, ) z = i, z = i, z 2 + dz = (2πi 0) = π. 2i (2) = ( i, ) z = i, z = i, z 2 + dz = (0 2πi) = π. 2i (3) = (0, 2) z = ±i, z 2 + dz = (2πi 2πi) = 0. 2i (4) = (, ) z = ±i, z 2 + dz = (0 0) = 0. 2i

121 = (0, 2) e z e z sin z () dz (2) z z 2 dz (3) (z π/2)(z + π) dz 0-2 n. = (0, 2) e z z + e iz () dz (2) z4 (z ) 2 dz (3) (z + 3) (3z π) 2 dz z 2 dz (z )(z + 2) () = (0, /2) (2) = (0, 3/2) (3) = (2, 3/2) (4) = (0, 3) 0-3. f(z) = (α, R) f(α) = 2π f(α + Re it ) dt 2π 0

122 () z = = (0, 2) e z z dz = 2πie = 2πei. e z (2) z 2 dz = e z (z )(z + ) dz ± = (±, /2) = e z z + e = 2πi + + = eπi. ez z e = 2πi = πi e. ( = e ) πi. e (3) sin z z + π sin z dz = 2πisin(π/2) (z π/2)(z + π) π/2 + π = 4i n. () z = 0 f(z) = e z 3 f (3) (0) = 3! e z e z 2πi dz. dz = 2πi z4 z4 3! e0 = πi 3. (2) z = 3 z = f(z) = z + z + 3 f () =! f(z) z + dz. 2πi (z ) 2 (z ) 2 (z + 3) dz = 2πi! f 2 () = 2πi (z + 3) 2 = πi z= 4. e iz (3) (3z π) 2 dz = e iz 9 (z π/3) 2 dz. z = π/3 f(z) = e iz ( π ) f =! e iz dz. 3 2πi (z π/3) 2 = 2πi 9 ( iei( π/3) ) = π( 3i) z 2 z = 0,, 2 (z )(z + 2) I := z 2 (z )(z + 2) dz () = (0, /2) z = 0 z =, 2 f(z) = (z )(z + 2) f (0) =! f(z) 2πi z 2 dz = ( ) 2z 2πi I I = 2πi f (0) = 2πi (z 2 + z 2) 2 = 2πi z=0 4 = πi 2. (2) = (0, 3/2) z = 0, z = 2

123 = (0, /3), = (, /3) = () πi 2 0 g(z) = z 2 (z + 2) = 2πi g() = 2πi. I = πi πi 3 = πi 6. (3) = (2, 3/2) z = z = 0, 2 (2) g(z) = z 2 (z + 2) I = = 2πi g() = 2πi 3. (4) = (0, 3) z = 0,, 2 2 = ( 2, /3) = h(z) = z 2 (z ) = 2πi h( 2) = ( 2 2πi ) = πi πi. (2) I = πi 6 = 0

124 i n () (2) n! n=0 n= -2. () z + 3 z n n 2 + z < (3) n=0 ( ) n z 2n+ (2n + )! z z = 0 z 4 (2) e z2 (3) cos z z -3. f(z) = z 2 α + () α = 0 (2) α = 2i (3) α = -. f(z) lim z z n f(z) n = 0 ϵ > 0 M > 0 z > M f(z)/z n < ϵ f(z)

125 () i n n! = n! = e < n=0 (2) z < z n n 2 + < n 2 + < n N. n2 n=0 (3) ( ) n z 2n+ (2n + )! = z 2n+ (2n + )! n N n N -2. () m 2N+ z + 3 = 3 z m m! n=0 n 2 < < e z <. + z/3 = ) z3 z2 ( z z4 3 4 = 3 z 9 + z2 27 z3 8 + z4. z < (2) e w = + w + w2 2! + w = z 2 e z2 = + ( z 2 ) + ( z2 ) 2 2! z 2 + z (3) cos z ( z = z2 z = (cos z) z 2 2 z3 2 3z4 +. z < 24 2! + z4 4! + = ) ( + z + z 2 + z 3 + z 4 + ) = z -3. () f(z) = z 2 + = ( z 2 ) n = ( ) n z 2n. n=0 n=0 (2) f(z) = (z i)(z + i) = ( 2i z i ). w = z 2i z + i z i = w + i = i + w/i = ( w ) n = (iw) n = i n w n ( w < ). i i i n=0 n=0 n=0 z + i = w + 3i = i n w n 3 n+ ( w < 3). w < n=0 z 2i < f(z) = ( ) ( ) i 2i 3 n+ i n w n n = 3 n+ 2 (z 2i)n. n=0 n=0 (3) (2) w = z z i = w + ( i) = i + w/( i) = ( ) n w n ( i) n+ ( w < i = 2). z + i = w + ( + i) = ( ) n w n ( + i) n+ ( w < +i = 2). w < 2 z < 2 f(z) = ( ) ( ) n ( )n ( ) 2i ( i) n+ ( + i) n+ w n = ( i) n+ ( ) n w n ( + i) n+. 2i n=0 w = z ±i = 2 e ±πi/4 n=0 n=0 n=0