I.2 z x, y i z = x + iy. x, y z (real part), (imaginary part), x = Re(z), y = Im(z). () i. (2) 2 z = x + iy, z 2 = x 2 + iy 2,, z ± z 2 = (x ± x 2 ) +

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1 I..... z 2 x, y z = x + iy (i ). 2 (x, y). 2.,,.,,. (), ( 2 ),,. II ( ).. z, w = f(z). z f(z), w. z = x + iy, f(z) 2 x, y. f(z) u(x, y), v(x, y), w = f(x + iy) = u(x, y) + iv(x, y).,. 2. z z, w w. D, D ( )

2 I.2 z x, y i z = x + iy. x, y z (real part), (imaginary part), x = Re(z), y = Im(z). () i. (2) 2 z = x + iy, z 2 = x 2 + iy 2,, z ± z 2 = (x ± x 2 ) + i(y ± y 2 ), z z 2 = (x x 2 y y 2 ) + i(x y 2 + x 2 y ),. z /z 2, x, y, x 2, y 2. (3) z = x+iy x iy z (complex conjugate), z.. (z z 2 ) = z z 2, z /z 2 = z /z 2 (z 2 ). (4) z = x + iy z x 2 + y 2. 2 z, z 2,. z + z 2 z + z 2 ( ) (5) n z, z 2,, z n,. z + z z n z + z z n (.) ( )

3 I 2.3 a, t. e at e at (at) n = n! n= = + at! + (at)2 + 2!.! =. e at, df(t) = af(t), f() =, dy. a. () e iy = cos y + i sin y (.2). y. (2) 2 z, z 2, e z e z 2 = e z +z 2 ( : e at ) ( )

4 I 3.4 z = x + iy (x, y ), (complex plane). z = r(cos θ + i sin θ) = re iθ (.3)... r = z = x 2 + y 2, θ z (argument), θ = argz. θ (phase) z /n : w n = z w z = re iθ z /n = n r exp[ i (θ + 2πk)] (k =,,... n ) n z /n n 8 /3 = 2, 2e 2π 3 i, 2e 4π 3 i () z, argz. (2) i, i, i, + 3i (3). (cos θ + i sin θ) n = cos nθ + i sin nθ, (.4) (4) z 3 =,. (5) n z n =. (6) /4, i 2/3, i i ( )

5 I 4 [7 ] z < z n = r < z n= r n cos nθ = n= r cos θ 2r cos θ + r 2 r n sin nθ = n= r sin θ 2r cos θ + r ( )

6 I 5.5 (.2), (.3). () cos(θ ± θ 2 ) = cos θ cos θ 2 sin θ sin θ 2 (2) sin(θ ± θ 2 ) = cos θ sin θ 2 ± sin θ cos θ 2 (3) cos 2θ = cos 2 θ sin 2 θ (4) sin 2θ = 2 sin θ cos θ ( )

7 I 6.6 k m x m m d2 x = kx (.5) dx2 d 2 x k dx = 2 ω2 x, ω 2 = (.6) m x = ae bt (.7) a, b b 2 ae bt = ω 2 ae bt b = ±ω x = a e iωt + a 2 e iωt (.8) t = x = A x() = A, ẋ() =. a, a 2 ẋ = iω(a e iωt a 2 e iωt ) a = a 2 = A/2 x = 2 A(eiωt + e iωt ) = A cos(ωt) (.9) () mẍ + 2λẋ + mω 2 x = λ λ (.9) ( )

8 I 7 2. D w = f(z). z D z, w w, f(z) z = z w. lim f(z) = w (2.) z z. z w w ( w < ) ( ). z z = re iθ r θ w = f(z) w f(z) z z w 2. w = (z ) 2 z z = re iθ r w = r 2 e 2iθ θ 2.2 lim (z z )2 = w = z z z z = re iθ w = r iθ re iθ = e 2iθ. r r w e 2iθ θ = w θ = π/2 w e iπ = w = f(z) 3. z = z f(z ) 2. lim z z f(z) = w 3. w = f(z ), f(z) z = z tex ()

9 I z 2 x, y z = x + iy f(z) x, y 2 f(z) = z 2 = x 2 y 2 + 2ixy f(z) = z 2 = x 2 + y 2. 2 x, y z z = x iy 2 z, z f = z 2 f(z) = z z z = x + iy, z = x iy. (2.2) x = 2 (z + z ), y = 2i (z z ). (2.3) f (x, y) (z, z ) f(x, y) z = z = x + iy x y f (z, z ) z = z ( ) ( ) f f f(z) = f(z ) + (z z ) + (z z z z z ) + O( z z 2 ) (2.4) z f(z) f(z ) z z = ( ) ( f f (z z ) + z z z )z + O( z z ) (2.5) z z z z z z = re iθ r ( ) ( ) f(z) f(z ) f f = (2.6) z z z z z + z e 2iθ ( f/ z ) z = z θ θ z ( ) f(z) z = z f (z ) df (z dz ) f (z ) z = z f z = (2.7) (2.7) - (Cauchy-Rieman) z z z tex ()

10 I 9 z 2 z 2 D f(z) f(z) D tex ()

11 I 2.3 (2.7) (2.7) / z x, y (2.3) x z = 2, y z = 2i z z x z = 2, z z = x z x + y z y z = 2i y = 2 ( x + i ) y (2.8) x, y z = ( 2 x ). (2.9) i y f(z) u v f(z) = u(x, y) + iv(x, y) (2.) (2.9) - (2.7) = x (u + iv) ( u (u + iv) = i y x v ) ( u y i y + v ) x u x = v y, v x = u y (2.) - (2.8). (2.9) 2 z z = ( 4 x ) ( i y x + ) = ( ) 2 i y 4 x + 2 =. (2.2) 2 y 2 4 (2.7) f f(z) = u(x, y) + i v(x, y) = u(x, y) = v(x, y) =. (2.3) tex ()

12 I z. e z, cos z, sin z.! =. e z = + z! + z2 2! + = z n n!, (2.4) n= cos z = z2 2! + z4 4! + = n= sin z = z z3 3! + z5 5! + = n= ( ) n z 2n, (2.5) (2n)! ( ) n z 2n+ (2n + )! (2.6) e iz = cos z + i sin z, e iz = cos z i sin z (2.7) 3 cos z = eiz + e iz, sin z = eiz e iz 2 2i (2.8) cos 2 z + sin 2 z = ( ) e cos 2 z + sin 2 iz + e iz 2 ( ) e iz e iz z = + = 2 2i d dz ez = e z, d d cos z = sin z, dz sin z = cos z (2.9) dz : cosh z = ez + e z, sinh z = ez e z 2 2, tanh z = sinh z cosh z (2.2) tex ()

13 I 2 z cosh 2 z sinh 2 z = (2.2) cosh z, sinh z (2.2) ( z < ) tanh z z = ±πi (2.2) z d d cosh z = sinh z, dz d sinh z = cosh z, dz dz tanh z = cosh 2 z (2.22) z = ±πi/2 cosh z = tanh z z = ±πi/2 3 cosh z z z z = re iθ ln z = ln r + iθ z ln 2 z (argument) i z ln z = ln z + i arg z (2.23) z > arg z = ln z = ln z ln( 2) = ln(2e iπ ) = ln 2 + iπ, ln i = π 2 i, ln( + i) = 2 ln 2 + iπ 4, (2.23) z = z = 2.7 z z θ = arg z = (2.23) 2 ln z ln z d dz ln z = z. (2.24) tex ()

14 I 3 z = z e iθ n z = z e i(θ+2nπ) = z e iθ e 2nπi. n e 2nπi = ln ln z = ln z + i(θ + 2nπ), n =, ±, ±2, (2.25) ln z z z z θ θ < 2π ln z z = re i(θ+2nπ), (n =, ±, ±2, ) f(z) = z = z /2 (2.26) z = re iθ z = re i(θ+2π) f(z) = z /2 f(re iθ ) = r /2 e iθ/2, f(re i(θ+2πi) ) = r /2 e iθ/2 e iπ = r /2 e iθ/2 f(re i(θ+4π) ) = r /2 e iθ/2 e 2πi = r /2 e iθ/2 = f(re iθ ) z r 2 (4π ) z /2 z /2 2 z z < 2π z /2 θ < π z /2 z = d dz z/2 = (2.27) 2z /2 z α α d dz zα = αz α (2.28) tex ()

15 I tex ()

16 I 5 2. f(z) x = (z + z )/2, y = (z z )/2i. f(z) = z z 2. f(z) = z + /z 3. f(z) = x 2 + y 2 + 2ixy, f(z) = x 2 + y 2 2ixy 4. f(z) = e y (cos x + i sin x), f(z) = e y (cos x i sin x) f(z) = u(x, y) + iv(x, y) u(x, y) v(x, y) V V = u f(z). df/dz = V x iv y 2. V = 3. V = z cosh z, sinh z, tanh z cosh z = cos iz, sinh z = i sin iz, tanh z = i tan iz cosh(z + z 2 ) = cosh z cosh z 2 + sinh z sinh z 2, sinh(z + z 2 ) = sinh z cosh z 2 + cosh z sinh z 2, tanh(z + z 2 ) = tanh z + tanh z 2 + tanh z tanh z 2 3? tex ()

17 I z = z = 2 (2.25). u(x, y) = ln z, v(x, y) = arg z u, v - (2.) 2. ln z z z - ln z/ z = (2.7) tex ()

18 I 7 3. dx x x x +i dz z + i () z = z = z = + i (2) z = i z = i z = + i () (2) z = x + iy (3.) x, y dz = dx + idy (3.2) +i dz z = +i (dx + idy)(x + iy) (3.3) () y = dy = + i x = dx = dz z = dx x + idy ( + iy) = dx x + i dy + i 2 dy y () = 2 + i + (3.4) i2 2 = i (2) i x = dx = i + i y = dy = (2) dz z = idy iy + () dx (x + i) = i 2 dy y + dx x + i dx = i (3.5) 2 + i = i tex ()

19 I 8 +i dz z (3.6) z = x iy (3.7) +i dz z = +i () dz z = dx x + idy ( iy) = (dx + idy)(x iy) (3.8) () dx x + i dy + i( i) dy y = 2 + i + 2 = + i (3.9) (2) dz z = idy ( iy) + dx (x i) = i( i) (2) dy y + dx x i dx = i = i (3.) () tex ()

20 I z = a z = b b a dz f(z) (3.) z = a b A z = b a B C C dz f(z) (3.2) C C f(z) z = x + iy f(z) x, y x + iy () (x + dx) + iy (2) (x + dx) + i(y + dy) (3) +i(y + dy) (4) x + iy. f(x, y)dx 2. f(x + dx, y)(idy) = if(x + dx, y)dy 3. f(x, y + dy)( dx) = f(x, y + dy)dx 4. f(x, y)( idy) = if(x, y)dy dz f(z) = [f(x, y) f(x, y + dy)]dx + i[f(x + dx, y) f(x, y)]dy ()+(2)+(3)+(4) = ( f ) ( ) f y + i f dxdy = i x x + i f dxdy y (3.3) C D D D C ( ) f dz f(z) = i dxdy x + i f (3.4) y C D (2.3) x = 2 (z + z ), y = 2i (z z ) tex ()

21 I 2 x, y z, z (x, y) idxdy = i (z, z ) dzdz = i 2 2i dzdz = 2 dzdz. (3.5) (2.9) C 2 2i x + i y = 2 (3.6) z dz f(z) = dzdz f (3.7) D z C f z = (3.8) dz f(z) = (3.9) (3.8) -f(z) D D f(z) f(z) +i [ z 2 dz z = 2 ] +i = ( + i)2 2 = i tex ()

22 I D z f(z) D f(z) z z z = z z r S r f(z)/(z z ) D S r D dz f(z) = dz f(z) (3.2) C z z S r z z S r z = z + re iθ (r = const, θ < 2π) (3.2) dz = ire iθ dθ (3.22) S r dz f(z) z z = 2π ire iθ dθ f(z + re iθ ) re iθ 2π = i dθ f(z + re iθ ). r integrand f(z ) = const 2πif(z ) f(z ) = dz f(z). (3.23) 2πi C z z f(z ) f(z) = const dz = 2πi (3.24) z z C tex ()

23 I f(z) D C D D z f(z ) (3.23) f(z ) = dz f(z) (3.25) 2πi C z z z f (z ) = 2πi f (z ) = 2 2πi C C dz dz f(z) (z z ) 2 (3.26) f(z) (z z ) 3 (3.27) f (z ) = 3 2 2πi f (n) (z ) = n! 2πi C C dz dz f(z) (z z ) 4. (3.28) f(z) (z z ) n+ (3.29) (3.29). f(z) z = z f (n) (z ) f(z) tex ()

24 I z = z z = z f(z) a f(z) a z z a > z a /(z z ) z = a z z = z a z a z a = k= (z a) k (z a) k+ (3.25) ( z a z a ) < (3.3) f(z ) = (z a) k dz f(z) 2πi C (z a) k+ k= = a k (z a) k. (3.3) k= z = a 2 a k = f(z) dz (3.32) 2πi (z a) k+ C z = a b b f(z) b f(z) a f(z) z tex ()

25 I f z = (3.33) f (z) f (n) tex ()

26 I i (3.) 3. i z z = re πi/4 ( r 2) 3.2 (3.2) 3.3 n C dz (z z ) n = { 2πi (n = ) (n ) C z 3.4 C z = 2 2πi dz z f(z) C dz f (z) f(z) = dz C z z C (z z ) 2 C 3.6 n. P n (x) = d n 2 n n! dx ( n x2 ) n P n (x) = 2 n 2πi dz ( z2 ) n (z x) n+ P (x) =, P (x) = x, P 2 (x) = 3 2 x2 2, tex ()

27 I H n (x) = ( ) n n! x2 e 2πi H n (x) = ( ) n e x2 dz dn dz n e x2 e z2 n! = (z x) n+ 2πi dz z n e z2 +2zx z z + x H (x) =, H (x) = 2x, H 2 (x) = 4x 2 2, H 3 (x) = 8x 3 2x, 3. L n (x) = ex 2πi L n (x) = ex d n n! dz n (xn e x ) dz z n e z (z x) = n+ 2πi ds e xs s ( s)s n+ z = x/( s) L (x) =, L (x) = x +, L 2 (x) = 2! (x2 4x + 2), tex ()

28 I f(z) D z a < z a f(z) a f(z ) = dz f(z) = a k (z a) k, a k = f(z) dz 2πi C z z 2πi C (z a) = f (k) (a) k+ k! k= D α f(z) α f(z) f(z) = /z 2 z /z 2 /z 2 z = f(z) = z 2 + z = ±i 2 f(z) = sin z z = nπ (n =, ±, ±2, ) f(z) = sin z z z = z = z sin z = ) (z z3 z z 3! + z = α f(z) f(z) (z α) k k lim (z z α α)k f(z) = a ( < a < ). (4.) lim(z i) z i z 2 + = lim(z i) z i (z + i)(z i) = 2i ()

29 I 28 (z α) k f(z) z α a α f(z) k (pole) z = i /(z 2 + ) f(z) = exp ( ) = z k= k! ( ) k z z = n z z n e /z f(z) ( f(z) = exp ) z z f(z) z = ()

30 I f(z) D z a < z a f(z) a f(z ) = dz f(z) = 2πi C z z a k = 2πi C dz a k (z a) k, (4.2) k= f(z) (z a) = f (k) (a) k+ k! (4.3) f(z) D α f(z) α α f(z) z = α f(z) z α f(z) (z α) k z α f(z) D α α D D D C D f(z) f(z) f(z ) = dz f(z) = dz f(z) dz f(z) (4.4) 2πi C z z 2πi C 2 z z 2πi C z z z D C 2, C C 2 z z α < z α C 2 = z z z α z α z α = k= (z α) k (z α) k+ (4.5) (4.4) dz f(z) = 2πi C 2 z z a k (z α) k (4.6) k= ()

31 I 3 a k = f(z) dz (4.7) 2πi C 2 (z α) k+ z z α < z α = z z z α (z α) k z α = (4.8) (z α) k k= z α (4.4) 2 dz f(z) = 2πi C z z a k = 2πi k= f(z) = a k (z α) k (4.9) dz f(z)(z α) k. (4.) k= a k (z α) k (4.) a k = f(z) dz (k =, ±, ±2, ) (4.2) 2πi C (z α) k+ C α D 3 f(z) D z = α f (k) (α) a k = f (k) (α) k! D f(z) α (4.2) k integrand D a k = (k =, 2, 3, ) k =,, 2,.. (4.2) f(z) z = α (k ) k < k = n (n > ) z = α n (pole) k = a z = α f(z) (residue) 3 C D D α ()

32 I f(z) z = α n f(z) f(z) = a n (z α) + a n+ n (z α) + a n+2 n (z α) + + a n 2 z α + a + a (z α) + (z α) n (z α) n f(z) = a n +a n+ (z α)+a n+2 (z α) 2 + +a (z α) n +a (z α) n +. n z a n,, a 2 d n dx [(z n α)n f(z)] = (n )!a + n!! a (z α) + z α a 4 Res f(α) a = (n + )! a (z α) ! [ ] d n (n )! dx {(z n α)n f(z)} z α (4.3) f(z) z = α n z Res f(z ) = [(z α)f(z)] z α (4.4) 4. f(z) = (z )(z + 2) z = z = 2 (4.4) (4.5) 4.2 Res f() = 3, Res f( ) = 3. f(z) = cos z z 3 z = 3 z = Res ( cos z z 3 )z= = 2! [ d 2 ( z 3 cos z = dz 2 z )]z= 3 2 [ cos z] z= = 2. cos z = ( z 3 z 3 2! z2 + ) 4! z4 = z 3 2z + 4! z 4 = residue = ()

33 I z = f(z) = exp ( ) z z z = f(z) = k= f (k) () z k (z ) k! f (k) () = lim z f (k) (z) 4.2 f(z) = (z )(z + 2) (4.6) z = (4.2) w = z /(z + 2) w w 4.3 f(z) = (z )(z + 2) (4.7) z = 4.4 z = π/2 f(z) = tan z 4.5 f(z) = sin z ()

34 I f(z) D f(z) = a k (z α) k = + a 2 (z α) + a 2 z α + a + a (z α) + (5.) k= z = α C dz f(z) = a k dz (z α) k. (5.2) C k= k = 3.3 k = 2πi a = Res f(α) dz f(z) = 2πi Res f(α) (5.3) C α k (k =, 2, ) α k C k C k D C C k f(z) dz f(z) = dz f(z) + dz f(z) + (5.4) C C C 2 C k C dz f(z) = 2πi k C k dz f(z) = 2πi Res f(α k ) (5.5) Res f(α k ) (5.6) C 2πi ()

35 I dx f(x) I = dx x 2 + d dx tan x = x 2 + (5.7) (5.8) dx x 2 + = [ tan x ] = π ( ) π 2 2 = π (5.9) C dz z 2 + integrand (5.) α = ±i (5.) C C. z = R z = R R C 2. R C 2 C C integrand α = i ( ) [ ] dz = 2πi Res = 2πi lim (z i) = π (5.2) z 2 + z 2 + z i z 2 + C z=i C 2 z = Re iθ C 2 R = const, < θ < π dz = ire iθ dθ dz π C z 2 + dθ R R 2 e 2iθ + = π dθ R + = π ( ) R + O (5.3) R 3 R 2 e 2iθ ()

36 I 35 R C C I R dz z 2 + = dx = π. (5.4) x 2 + C f(z). 2. arg z π z f(z) /z 2 (f(z) = O(/z 2 ) for z ) dx f(x) = 2πi ( ) (5.5) ()

37 I dx f(x)e iax dx sin x x z dz eiz z (5.6) (5.7) R > r >. R < z < r 2. C r r (π > θ > ) 3. r < z < R 4. C R R ( < θ < π) integrand r R dz eiz z = = dx eix R x Cr + dz eiz z + dx eix r x CR + dz eiz z (5.8) C R z = Re iθ = R(cos θ + i sin θ), dz = ire iθ dθ ( < θ < π) (5.9) CR dz eiz z = i π π dθ e R(i cos θ sin θ) = i dθ e R sin θ e ir cos θ (5.2) < θ < π sin θ > R integrand R CR dz eiz z (R ) (5.2) C r Cr dz eiz z = i π dθ e r(i cos θ sin θ) (5.22) ()

38 I 37 r e r(i cos θ sin θ) Cr dz eiz z iπ (r ). (5.23) R r r R r dx eix x + dx eix R x = dx eix r x + dx e ix R x r, R R = 2i dx sin x r x. (5.24) dx sin x x = π 2 (5.25) f(z). 2. z arg z π f(z) lim f(z) = ( arg z π) (5.26) z dx f(x)e iax = 2πi ( ) (a > ) (5.27) ()

39 I 38 2π 5.4 dθ f(cos θ, sin θ) 2π z = e iθ 2π dθ a + b cos θ dθ a + b cos θ = 2 i (a > b > ) (5.28) dz bz 2 + 2az + b Integrand (5.29) z = b ( a ± a 2 b 2 ) = { z+ z (5.3) z + z = z + ( ) ( ) Res = Res bz 2 + 2az + b z=z + b(z z + )(z z ) z=z + (5.3) = b(z + z ) = 2 a 2 b. 2 dz bz 2 + 2az + b ( ) = 2πi Res bz 2 + 2az + b z=z + = πi a2 b 2. (5.32) 2π dθ a + b cos θ = 2π a2 b 2 (a > b > ). (5.33) 2π z = e iθ 2π dθ = i dθ f(cos θ, sin θ) (5.34) dz z, cos θ = 2 (z + z ), sin θ = 2i (z z ) (5.35) dθ f(cos θ, sin θ) = i ( dz z f 2 (z + z ), ) 2i (z z ) (5.36) ()

40 I 39 f(z)/z z n (n =, 2, ) 2π dθ f(cos θ, sin θ) = 2π ( ) f(z) Res (5.37) z n z m f(z)/z ()

41 I dx x 2 + C dx dx x 4 + x 2 x 4 + dx (x 2 + a 2 ) = π (a > ) 2 4a dx sin x x C r r (π < θ < 2π) 5.6 a I(a) = dx e ax2. I(a) a 2. a dx e ax2 = 2 π a (5.38) ()

42 I (Fresnel) dx cos x 2, 5.8 I = 2π t > t = dx sin x 2 dθ + t 2 2t cos θ = 2π t 2 ( t < ) P r r P r (r < r) P P l l 2 = (r r ) 2 = r 2 ( + t 2 2t cos θ) t = r /r, θ r r I r r P P l 2 = l 2 r 2 r 2 P 5.9 a > (a) dx cos x x 2 + a 2 = π a e a. cos x cos kx (b) dx x sin x x 2 + a 2 = πe a. sin x sin kx 5. z = z f(z) f(z) = a k (z z ) k, a k = f(z) dz 2πi (z z ) k+ k= (4.2) f(z) = (z )(z + 2) z = ( z < 3) a k (??) ()

43 I f(z) = (z )(z + 2) z > 3 z z = (??) ()

44 I dx f(x). x x f(x ) 2. x δ x 2. x ɛ x x +iɛ x iɛ (ɛ ) f(z) z. < x < 2. R () + (2) x C δ+ C δ δ C δ± dz f(z) z x = { iπf(x ) C δ+ iπf(x ) C δ C δ+ C dz f(z) x δ = z x + dx f(x) + dz f(z) + dx f(x) x x C δ± z x x +δ x x C R dz f(z) z x (δ, R ). (6.) f(z) (2) R (2) ()

45 I 44 C δ+ C integrand z = x f(x ) 2πif(x ) 2 (6.) iπf(x ) C δ C integrand 2 iπf(x ) f(z) z ( arg z π) P dx f(x) x x iπf(x ) = (6.2) P dx f(x) x x = lim δ ( x δ dx f(x) + dx f(x) ) x x x +δ x x (δ ) (6.3) (principal value) δ f(z) z ( π < arg z < 2π) P dx f(x) x x + iπf(x ) = (6.4) < x < a x +δ b a dx x x (a < x < b) δ, δ ( x δ ) b dx + dx = [ln( δ ) ln(a x )] + [ln(b x ) ln δ] x x = ln b x + ln δ a x δ δ δ δ = δ ln( δ /δ) = ln( ) = iπ ( x δ a dx + b x +δ ) b dx dx = P = ln b x + iπ x x a x x a x ()

46 I 45 7 ± P a dx f(x) lim dx f(x) (6.5) a a 7 ln z arg z < 2π ()

47 I dx f(x) x x. 2. f(z) z C. < x < 2. R () + (2) (2) (5.3) x x ± iɛ (ɛ ) { 2πif(x ) (x x + iɛ) C dz ( 2 f(z) z x iɛ = dx dx f(x) x x iɛ + f(x) x x iɛ = dx (x x iɛ) (6.6) ) f(x) = iπf(z) = P dx f(x). x x + iɛ x x (6.7) x x + iɛ x iɛ (6.2) (6.6) dx f(x) x x iɛ = P dx f(x) ± iπf(x ) (6.8) x x f(z) z x x iɛ = P ± iπδ(x x ) (6.9) x x δ(x x ) b a dx f(x)δ(x x ) = f(x ) (a < x < b), (6.) dx δ(x x ) =, δ(x) = (x ) (6.) ()

48 I 47 (6.9) dx f(x) (6.9) δ δ(x x ) = ( ) 2πi x x iɛ = x x + iɛ π ɛ (6.2) (x x ) 2 + ɛ 2 δ dx δ(x x ) = π δ(x x ) = (x x, ɛ ), dx ɛ (x x ) 2 + ɛ 2 = π dy y 2 + = (ɛy = x x ). dx x 2 + = π (5.4) ()

49 I dx sin x x P dx eix x z f(z) = e iz (6.8) C dz e iz z iɛ = dx eix x iɛ = P dx eix x ± iπ (6.3) C () + (2) R + integrand z = iɛ C e iz /(z + iɛ) P dx eix x iπ = (6.4) dx sin x x = π (6.5) (6.3) ()

50 I I(σ) = dx x sin x x 2 σ 2 (6.6) x σ I - ( + k 2 )G(r, r ) = δ 3 (r r ) G(r, r ) I(σ) x = σ 6.. (6.6) I(σ) I(σ) e iσ sin z = 2i (eiz e iz ) (6.6) I (σ) = 2i I 2 (σ) = 2i dz I(σ) = I (σ) + I 2 (σ), (6.7) ze iz z 2 σ = 2 2i dz ze iz z 2 σ 2 = 2i dz (z + σ)(z σ), (6.8) dz ze iz ze iz (z + σ)(z σ) (6.9) I I 2 I, I 2 z = ±σ Res I ( σ) = e iσ 2, Res I (σ) = eiσ 2, Res I 2 ( σ) = eiσ 2, Res I 2(σ) = e iσ 2 (6.2) ()

51 I I, I 2, I z = ±σ I I 2 P I iπ 2i P I 2 + iπ 2i ( ) e iσ 2 ( ) e iσ 2 iπ 2i + iπ 2i ( ) e iσ =. (6.2) 2 ( ) e iσ =. (6.22) 2 P I = P I + P I 2 = π 2 (eiσ + e iσ ) = π cos σ (6.23) 2 σ σ + iɛ (6.24) I + (σ + iɛ) = 2πi ( ) e i(σ+iɛ) = π 2i 2 2 eiσ, (ɛ ) I (σ + iɛ) = 2πi ( ) e i(σ+iɛ) = π 2i 2 2 eiσ (ɛ ) (6.25) I(σ) = I + + I = πe iσ (ɛ ) (6.26) σ σ iɛ (6.27) I(σ) = πe iσ (6.28) (6.23) (6.23) ()

52 I 5 6. (6.4) 6.2 θ(s) θ(s) = { (s < ) (s > ) ɛ > θ(s) (a) u(s) = dx 2πi (b) u(s) = 2 + 2πi P eixs x iɛ dx eixs x 6.3 I = dx x 2 σ 2 (a) (b) (c) σ σ + iɛ σ σ iɛ ()

53 I (saddle point method) (method of steepest decent) I(s) = dz g(z)e sf(z) (7.) C s I(s) s C C f(z) C e sf(z). (7.2) g(z) e sf(z) ()

54 I n n! s s! s! = dx x s e x (7.3) 8 s s = n n! (7.3) ( )! = dx x /2 e x. 2 y = x /2, x = y 2, dx = 2ydy ( ) ( d )! = 2 dy y 2 e y2 = 2 dy e ay2 2 da a= [ ( )] d π π = 2 = da 2 a a= 2 dy e ay2 = 2 π a (5.38) s I(s) (7.3) (7.) x = sz, dx = sdz (7.3) s! = s s+ dz z s e s = s s+ dz e sf(z) (7.4) C (7.) C f(z) = ln z z, g(z) = (7.5) f(z) integrand f (z) = z, f (z) = z 2 8 s! Γ(s + ) = s! (/2)! = Γ(3/2) ()

55 I 54 z = z = f(z) z = re iθ f(z) = f() + 2! f ()(z ) 2 + = 2 (z )2 + = r2 2 e2iθ +, e sf(z) = e s exp [ s ] 2 r2 e 2iθ z = (r = ) e sf(z) θ = θ = π/2 z = θ = π s θ = θ =, π r integrand e sf(z) (7.4) r = r = (7.4) θ = s! s s+ e s dr exp ( s ) 2π 2 r2 = s s+ e s s dx e ax2 = π a (5.38) s! 2πss s e s (7.6) s (7.6) (Stiring) s s e s N ln ln N! N ln N N (7.7) N ln N N ln N s! e s s e s (e s /s! for s ) ()

56 I H ν () (s) H ν () (s) = iπ C H ν (2) (s) = dz e s iπ C 2 dz e s 2(z z), (7.8) z ν+ 2(z z) z ν+. (7.9) C C z = +i C 2 C i J ν (s) N ν (s) J ν (s) N ν (s) H () ν (s) = J ν (s) + in ν (s), (7.) H (2) ν (s) = J ν (s) in ν (s), (7.) H ν () (s), H ν (2) (s) 2 9 H ν () (s) s s H ν () (s) s (7.8) f(z) = ( z ), g(z) = 2 z z ν+ H ν () (s) (7.) z = z = Rf(z) = f(z) f (z) = f(z) f (z) = ( + z ), f (z) = 2 2 z 3 f (z) z = ±i C z = +i z = i f(z) f(z) = f(i) + 2 f (i)(z i) 2 + i i 2 r2 e 2iθ 9 ν ()

57 I 56 z i = re iθ e 2(z s z) = e sf(z) e i exp ( is2 ) r2 e 2iθ. z = i (r = ) θ r = exp ( is2 ) r2 e 2iθ θ θ = 3π/4 θ = π/4 exp ( is2 ) r2 e 2iθ = exp ( 2 ) sr2 r = s s r = g(z) g(z) g(i) = /i ν+ dz = dr e iθ dr e 3πi/4 r H ν () (s) iπ e3πi/4 e is dr exp ( 2 ) i ν+ sr2 = iπ e3πi/4 e is 2π i ν+ s dx e ax2 = π a (5.38) H ν () (s) H ν () (s) 2 [ ( πs exp i s (2ν + ) π )] 4 (7.2) θ = π/4 exp( isr 2 e 2iθ /2) r = r = (i.e. z = i) exp( isr 2 e 2iθ /2) ()

58 I s I(s) = dz g(z)e sf(z) C e sf(z). f (z) = z f (z ) =. 2. z z = re iθ z f(z) sf(z) = s [f(z ) + 2 ] f (z )(z z ) 2 = sf(z ) + 2 r2 sf (z )e 2iθ C 3. sf (z )e 2iθ = sf (z ) θ 4. θ sf (z ) z z = z g(z) g(z ) r = dr exp ( 2 ) sf (z ) r 2 = 2π sf (z ) 5. 2π I(s) sf (z ) g(z )e sf(z) e iθ ( sf (z ) ) (7.3) s (7.3) θ ()

59 I s s! (7.6) (7.7) s! N! s N 7.2 s s! 2π s ss+ e s ( s ) 7.3 s! = s! = dx x s e x (7.3) s ln x x dx e f(z) = s ln z z H ν (2) (s) 2 [ ( πs exp i s (2ν + ) π )] 4 (7.4) ()

60 I f(z) f(z) α n ( < α < α 2 < ) α n f(z) b n f(z) f(z) z = α n (n =, 2, ) b n f(z) b n f(z) z α n n= z = f() z = f() f(z) = f() + n= ( b n + ) z α n α n (8.) (8.) R n C n C n α, α 2,, α n α n+ n R n C n I n = dw f(w) 2πi C n w(w z) = ( dw f(w) 2πiz C n w z ) (8.2) w n R n I n < 2π max 2πR n f(z) z C n R n (R n z ) = max f(z) z C n R n (R n z ). (8.3) z f(z) /R n ε i.e. f(z) z f(z) = o(z) for z ε I n < (8.4) R n z n I n (8.2) w = z w = z f(z) f(z) f(z)/(w z) b k /(α k z) 2πi dw f(w) n C n w z = f(z) + Res k= z = f(z) ( f(w) w z ) w=α k = f(z) + k= b k α k z. (8.5) ()

61 I 6 2 dw f(w) 2πi C n w n ( ) f(w) = f() Res w k= I n = f(z) f() + n k= n I n (8.) w=α k = f() b k α k. (8.6) ( b k a k z ). (8.7) α k 8. cot z = cos z sin z z = (8.8) f(z) = z cot z f() = lim z f(z) = f(z) a n = nπ (n = ±, ±2, ) b n = nπ (8.) f(z) = + n= ( nπ z nπ + ). nπ n = n n= ( nπ z nπ + ) = 2z 2 nπ n= f(z) = + 2z 2 n= cot z = z + 2z n= z 2 n 2 π 2 z 2 n 2 π 2 z 2 n 2 π 2 (8.9) ()

62 I (8.9) cot z = d ln sin z dz (8.9) ln sin z = ln z + ln(z 2 n 2 π 2 ) + C = ln z + 2 n= sin z z = c ( z2 n= n 2 π 2 n= ). ) ln ( z2 + C n 2 π 2 z (sin z)/z c c = ) sin z = z ( z2 (8.) n 2 π 2 n= sin z f(z) z ( z < ) a n (n = ±, ±2, ) a n f(z) = (z a n )g(z) g(z) g(a n ) f(z) f (z) f(z) = + g (z) z a n g(z) (8.) z = a n g /g z = a n f /f f (z) f(z) = f () f() + ( + ) (8.2) z a n a n z dz f (z) f(z) = ln f(z) ln f() = zf () f() + f(z) = f() exp n= n= [ln(z a n ) ln( a n ) + zan ]. [ ] zf () ) ( ) ( zan z exp f() a n= n (8.3) (8.4) 2 ln(z 2 n 2 π 2 ) (8.9) ()

63 I sin z sin z = z + 2z ( ) n z 2 n 2 π 2 z= f(z) = z/ sin z f() 8.2 cos z cos z = n= n= [ ] z 2 (n /2)π ()

64 I ζ ζ ζ(s) = n= n s (9.) ζ(4) ζ(4) cot πz dz (9.2) z 4 N (N + /2, N /2), (N + /2, N + /2), ( N /2, N + /2), ( N /2, N /2) 4 N lim N cot πz dz =. (9.3) z 4 n z = n ( ) cot πz Res =. (9.4) n= n ( ) cot πz Res (9.4) = = 2 π n= n= z 4 z=n ( ) cot πz Res z 4 n 4 + Res ζ(4) = n= z 4 z=n = lim z n (z n) cos πz z 4 sin πz z=n = 2 ( ) cot πz z 4 z= n 4 = π 2 Res ( ) cot πz Res n= ( ) cot πz z 4 = πn 4. (9.5) z 4 z= z= (9.6) ζ(4) (9.2) z = z = 5 ( ) cot πz Res = ( ) ( ) d 4 cot πz z 4 z5 = π3 d 4 z= 4! dz4 z 4 z 24 dz z cot z (9.7) 4 z ζ(4) = π4 48 ( ) d 4 dz z cot z 4 z = π4 9. (9.8) ζ ()

65 I d 4 (z cot z) dz4 = 4(4 cot 2 z cosec 2 z + 2cosec 4 z) + z(8 cot 3 z cosec 2 z + 6 cot z cosec 4 z). lim z d 4 8 (z cot z) = dz4 5. res = π3 45. ζ ()

66 I k ζ(k) ζ(k) (k: even) 2πi cot πz dz = = 2 z k ( ) cot πz Res n= z k z=n ( ) cot πz + Res z k z= (9.9) ( ) cot πz Res = z 4 z= k! ( ) cot πz Res ( d k z k+ dz k z k cot πz z k = πn, (9.) ) k = πk d k (ξ cot ξ) k! dzk (9.) z z=n z ζ(k) = n= n k = πk 2k! d k dz (ξ cot ξ) k. (9.2) z ζ(2) = π2 4 d 2 (ξ cot ξ) dz2 = π2 z 6. (9.3) ζ ()

67 I n= ( )n /n k (k: even) 2πi dz z k sin πz (9.4) dz 2πi z k sin πz = = ( ) dz Res. (9.5) z k sin πz z=n Res ( dz ) z k sin πz ( ) dz Res z k sin πz d k = z= k! dz k z=n ( z k+ ( ) n n= n k = ( )n (n ), (9.6) πn k ) ( ) = πk d k ξ z k sin πz k! dz k sin ξ z. z (9.7) ( ) = πk d k ξ. (9.8) k! dz k sin ξ z n = k k 2 n= ( )n /n k ( ) n n= n k = πk d k 2k! dξ k ( ) ξ sin ξ z (k = even). (9.9) k = 2 ξ ( ) n n 2 n= = π2 4 ( ) d 2 ξ. (9.2) dξ 2 sin ξ z ξ sin ξ = ξ ( ξ ξ 3 /3! + ξ 5 /5! = 6 ) ξ2 + ξ4 2 = + 6 ξ2 + O(ξ 4 ). ( ) d 2 ξ = dξ 2 sin ξ z 3. ( ) n n= n 2 = π4 2. (9.2) ζ ()

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