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1 B ( ) :,..,,., f(z): D C: D a C, 2πi C f(z) dz = f(a). z a a C, ( ). (ii), a D, a U a,r D f. f(z) = A n (z a) n, z U a,r, n= A n := 2πi C f(ζ) dζ, n =,,..., (ζ a) n+, C a D. (iii) U a,r D f f (n) (a) = n!a n, n =,,..., ( ). f (n) (a) n! = 2πi C f(ζ) dζ, n =,,.... (ζ a) n+ 5.., J J62,

2 (ii) (iii) z a = z a =2 z a =6 sin z z a dz. cosh z z a dz. e z dz. (, a b > 6.) (z a)(z b) 52.. (a) dz, ( a = /6.), (b) z =6 az (c) z a =2 53. f n (z) := z a = sin z (z a) 3 dz, e z dz. (, < a b < 2.) (z a)(z b) n j= z j = + z + + z n, f(z) :=. < r <, z C r := {z C : z = r}, f n C r f. z, lim n f n (z ) = f(z ). (ii) f g C := sup z C r f(z) g(z). ( h C h. f g C C r.) f f n C rn+ r. (iii) f n f C r., f n f A n, sup f n (z) f(z) z A., R U,R f(z), f(z) M, z U,R., f (n) Mn! (z), n =, 2,.... (z =, (R z ) n ( ).) (ii) f ( ),.. (.) (iii) f(z) n, n, α,..., α n, f(z) = c(z α )(z α 2 ) (z α n ), c. (.) 2

3 f(z) z =, R > f (n) () f(z) = z n, z < R,., n! n=. 56. Q n (z) n. Q n (z) = z = α. ( : /Q n (z).) (ii) z = α Q n (z), Q n (z) = (z α)r n (z), R n (z) n,. 57. n Q n (z) = a + a z + + a n z n + z n. α,..., α n. Q n(z) Q n (z). Q n(z)/q n (z) = (ii), R C R, I R := Q n(z) 2πi C R Q n (z) dz n j=., R > max{ α,..., α n }. (iii) lim R I R. (.) z α j. 4.2 D f, z = c f r > s.t. < z c < r f. c / D. ( : f(z) = /(z c) D = C \ {c}, f z = c.) 58. cos z tan z := sin z cos z.. tan z tan /z, z =. (ii), z =. 3

4 f < z c < R D. (, z = c.) D r z c r 2 f(z) = n= a n (z c) n.,. (ii) a (residue). Res z=a f := a (= f(z) dz ) 2πi (iii) n= z c =ϵ a n (z c) n., z = c., a k k, z = c., a k (z c) k + + a z c, a k, z = c k. 59. f(z) = sin z z. z =. (ii) z =. 6. f(z) = sin2 z z 2. z =. (ii) z =. ( : sin 2 z cos 2z.) 6. f(z) = z 2 (z + 3). z = 3, < z + 3 < 3. (ii) z =, < z < 3. 4

5 62. f(z) = e /z. z =. (ii) z =. 63. f(z) = e /z. < r < r 2 <. 62, r z r 2 n, : f n (z) := a j z sup f(z) j j= r z r 2 f n (z) a j r j. j=n+ (ii) < r 2 <. 62, < z r 2 n. : f n (z) := a j z sup f(z) j < z r 2 f n (z) =. j= f < r < z c < R D. (, z = c.) D r z c r 2 f(z) = n= a n (z c) n. 64. z 2 (z 2). (a) < z < 2 z =. (b) 2 < z < z =. (a). 65. z =, z =,,., < z <. z(z 2 + ) (ii) sin z z. (iii) z e z (.) 5

6 z 66. z = < z < 2. z 2 3z + 2 f, C z,..., z n. (, z = z,..., z n f.),. C f(z) dz 2πi = n Res z=zj f, f z = c k, (z c) k f(z) = a k + a k+ (z c) +.,. j= 67. z =ϵ 68. f(z) = dz sin z 2πi a = ( ) k d (k )! lim { (z c) k f(z) }. z c dz., ϵ. z 2 (z + 3). (a). (b) (a). (c) (a). sin z sin(z ) 69. f(z) =. z =, z =, z 2 (z ) 3. ( : z = f(z) = a 2 + a +... (z ) 2 z., a 2.) 4.3 w := /z, w = z = ( ). f(z) = f(/w) = g(w) z = w = g(w). z = w = g(w). k. 6

7 7. w = /z. (a) z < w. (b) z = re iθ, θ 2π w.. (c) e /z z =,,. 7. f(z) = e /z,. z =. (ii) z n = /n, lim n f(z n ). (iii) z n = /n, lim n f(z n ). 72. f(z) = e /z,., k >, δ < 2π. r n := 2nπ δ, θ n := π 2 kr n, n =, 2,...,, lim n cos θ n r n (ii) lim e i ( sin θn ) rn = e iδ. n = k. (iii) α( ) lim n z n =, lim n e /z n = α {z n }. z = a α ( + ) lim z n =, lim f(z n ) = α n n {z n }.. 73., f(z) z = c. ( ) z = c lim z c f(z). (ii) z = c k lim z c f(z) = +. (iii) z = c lim z c f(z). 7

8 5 x 2 + arctan x,. dx x 2 + C R R(> ). f(z) z = R /R 2 C R e iaz f(z)dz, R. ( ). arg z π z, f(z), a >, I R := e iaz f(z)dz, R C R. ( )θ arg z θ 2 ( π). 74. f(z) z > R >, f(z) A. z 2,. I R := e iaz f(z)dz, R C R, C R, R(> R ). 75. J R := CR e iaz (z i) 2 dz., C R, R, a,. (a) z = Re iθ = R(cos θ + i sin θ), θ π,. J R π ar sin θ e (R ) dθ 2 (b) a lim R J R., a <. 76. Q n (z) z n (n ). 8

9 (a) R., A >, z R Q n (z) A R n. ( Q n (z) = O(R n ). ) (b) n 2. a., J R :=. CR e iaz dz lim Q n (z) J R R g(x) Q n (x) dx, Q n(x),, x g(x)/q n (x). 77. m P m (x), n Q n (x), n 2 m P m (x) Q n (x) dx., Q n (x).,. P m (z) lim R C R Q n (z) dz =., C R, R. (ii), P m (x) k Q n (x) dx = 2πi P m (z) Res z=αj Q n (z)., α,..., α k Imz > k (ii) (x 2 + ) dx. x 2 (x 2 + )(x 2 + 3) dx ( ) cos ax x 2 + dx, x sin ax dx, a >. x j=

10 (ii) (iii) (iv) cos x x 4 + dx, + x 6 dx. x sin ax dx, a >. x 4 + ax 4 + 2bx 2 + c dx, a, b, c b2 > ac. sin x, dx x = x, /z. 8.. sin x x dx x a (a > ), log x. a, z = Re iθ, θ 2π z a z a = R a (Re i2π ) a = R a e i2πa x a + x dx, < a < θ 2π z(θ) := e iθ, θ 2π cos θ = (z + /z)/2, sin θ = (z /z)/2i. ( z = z = /z.) 82.., a >. (ii) (iii) 2π 2π 2π 2a cos θ + a 2 dθ. sin 2 θ 2a cos θ + a 2 dθ. sin 2 θ + a cos θ dθ.

11 83.. (ii) 2π 2π + a cos θ dθ = 2π, < a <. a 2 ( + a cos θ) dθ = 2π 2 ( a 2 ), < a <. 3,.,. 84 ( ).. (a) I = e x2 dx = { 2π } e y2 dy, I 2 = re r2 dθ dr. (b) a > e ax2 /2 dx = 2πa /2. (c) (b) a a = 2π x 2m e x2 /2 dx, m =,, 2,... m. (.) 85 ( ).. a, b, c,. (a) (b) 2π 2π e (ax2 +2bx+c)/2 dx, a >. cosh x e x2 /2 dx. 86 ( ). I =. (a) t. it J = (b) (c) e x2 2 dx +it +it e z2 2 dz. R, R 2, R 2 + it, R + it, R, R 2 J = I. ( : K(R) :=, R K(R).) 2π 2π e itx e x2 /2 dx (t R). cos x e x2 /2 dx. t e 2 (R+iy)2 idy

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s ... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z

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n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt

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x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x [ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),

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z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =

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