b n c n d n d n = f() d (n =, ±, ±, ) () πi ( a) n+ () () = a R a f() = a k Γ ( < k < R) Γ f() Γ ζ R ζ k a Γ f() = f(ζ) πi ζ dζ f(ζ) dζ (3) πi Γ ζ (3)

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Download "b n c n d n d n = f() d (n =, ±, ±, ) () πi ( a) n+ () () = a R a f() = a k Γ ( < k < R) Γ f() Γ ζ R ζ k a Γ f() = f(ζ) πi ζ dζ f(ζ) dζ (3) πi Γ ζ (3)"

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1 [ ] KENZOU Origin 6//5) 3 a a f() = b n ( a) n c n + ( a) n n= n= = b + b ( a) + b ( a) + + c a + c ( a) + b n = f() πi ( a) n+ d, c n = f() d πi ( a) n+ ()

2 b n c n d n d n = f() d (n =, ±, ±, ) () πi ( a) n+ () () = a R a f() = a k Γ ( < k < R) Γ f() Γ ζ R ζ k a Γ f() = f(ζ) πi ζ dζ f(ζ) dζ (3) πi Γ ζ (3) ζ a ζ a < ζ ζ = (ζ a) ( a) = ( a ) = ( ) a n (4) ζ a ζ a ζ a ζ a ζ Γ Γ ζ a a < ζ ζ = ( a) (ζ a) = ( ζ a a a ) = a n= n= ( ) ζ a n (5) a

3 f() = ( a) n πi n= f(ζ) (ζ a) n+ dζ + ( a) n (ζ a) n f(ζ)dζ (6) πi n= Γ (5) N = n N n ( a) n (ζ a) n f(ζ)dζ = πi n= Γ N= n= ( a) N πi ( a) n πi Γ Γ f(ζ) dζ (ζ a) N+ f(ζ) dζ (ζ a) n+ f() Γ Γ (7) D f() D, f()d = f()d D (6) f() = n= c n ( a) n, c n = πi f() d (n =, ±, ±, ) (8) ( a) n+. (4)(5) (6) a < R a > k ➀ a < R ➁ a > k ➂ k < R 3

4 ➂ (8) (8) f() = c n ( a) n + n= n= c n ( a) n = ϕ() + n= c n ( a) n (9) ϕ() = c + c ( a) + c ( a) + + c n ( a) n + ϕ() = a c n (9) ( a) n ( n=. = f() = ( + ) (8) c n = = ±i ( ) = ➀ ϵ r < ϵ < r < ) < < = ( ) ➁ = ±i > f() = ( ) ➀ < < < + = ( ) + ( ) ( ) 3 + = + = ( + ) = + ( ) n ( ) n = + ( ) n+ n+ ➁ > / < Γ 4 n= n=

5 > i < - ϵ i = + / = ( + ) = + ( ) n n = ( ) n+ n= n= n+ / f() = + = i f() = + = ±i ( i)( + i) i i i i < i < < i f() /( i) i ➀ < i < ( i)/i < 5

6 i + i = i i + i = i i[ + ( i)/i] [ = { } ( i) { } ] ( i) 3 + i( i) i i i = i( i) (i) + i ( i) (i) 3 (i) 4 ➁ i > i = i Γ i/( i) < i + i = i = = i + i = i [ ( i) i ( i) + { } ( i) + i ( i) { } i { } ] i 3 ( i) ( i) ( i) i ( i) 3 + (i) ( i) 4 (i)3 ( i) 5 ( f() f() = + = [ + i ] ( ) = n + ( < ) ( + ) m ( m)( m ) = m + +! ( m)( m ) ( m n + ) + n + ( < ) n! e = n + ( ) n! Log( + ) = ( )n+ n + ( < ) n sin = 3 3! 5 5! + + ( )n (n + )! n+ + ( ) cos =! + 4 4! + ( )n (n)! n + ( ) () () () 3 = () e / () sin 6

7 () > e / () sin 4 e () ( = ) () ( = i) (3) ( i) ( ) ( = ) () f() = ( )( + ) (3) < < + < < < + < i 4-4- < < /( ) () f() = + = ( ( ) = + ) ( + ) + = (4) f() = n= n= ( ( ) n = ) (4) n= ( ) n ( ) n (5) ( ) n+ ( ) n (6) > < f() f() = + = ( ) ( = ) n = n= n= 7 ( + ) ( ) n ( ) n+ (7)

8 () i < i > /( i) f() = ( i) = i = i i + i) = i i + i i [ = i i + + i( i) + in ( i) n + = i ( ) ] i n (8) ( ) n i i ( + ) > n= f() = = ( i) = i i + i) = i ( i) + i i = ( i) i + i ( ) i n (9) ( ) n i (3) = ( ) = e e f() = ( ) = ( ) e { ( )e = ( ) e + + ( ) e!! [ ] e ( ) n = ( ) n! n= n= + + ( )n e n! } + 5 a + + = i + ±i + i i a i () a i a + i ➀ a < a ± i 8

9 a a ± i < + = i + + i = (a i) + ( a) + (a + i) + ( a) = (a i)( + a a i ) + (a + i)( + a a+i ) = ( ) a n ( ) n + ( a ( ) n a i a i a + i a + i n= n= [ (a + i) = ( ) n ( a) n n+ + (a i) n+ ] [(a i) (a + i)] n+ ] n= = ( ) n (a + i)n+ + (a i) n+ (a + ) n+ ( a) n n= ➁ a > a ± i Γ a ± i a < + = ( a)( + a i a ) + ( a)( + a+i = ( ) n (a i)n + (a + i) n ( a) n+ n= ( a ) 6 f() = ( )( i) = i f() = + i 5 ➀ < < i = i ( ) i ➂ < i = i 9 ( ) i ) n ➁ < < = ( )

10 = i < < ( ) < < ( ) < ( ) = ➀ f() = + i 5 ➂ f() = + i 5 ( ) (i) k+ n, (i) n n= n= n ➁ f() = + i 5 ( n= ) infty n + k (i) k+ n= offee Break : ***** T ea T ime ***** f() f() f() = c n ( a) n, c n = πi f() d (n =, ±, ±, ) () ( a) n+ a f()d = c n ( a) n d () a = re iθ, ( a) n = r n e inθ d, dθ = ireiθ π π f()d = c n r n e inθ ire iθ dθ = i c n r n+ e i(n+)θ dθ n= n= (3)

11 a θ n = n + π π ic n r n+ e i(n+)θ dθ = ic dθ = πic (4) π [ ic n r n+ e i(n+)θ dθ = ic n r n+ (n + ) ei(n+)θ ] π = (5) (3) n = ( a) f()d = πic (6) f() (6) c a f() a f() Res[f, a] Res[a] a f() f() = ϕ() + c a + c ( a) + + c k ( a) k (7) ϕ() a Res[f, a] = f()d = c (8) πi n = ( f()d = d = πi, a (9) ( a) n d = (n > ) c n (9) f() = ϕ() + ϕ() ϕ()d = ( a) n n= c n c n d = πi ( a) n n=

12 . Singilar Point) f() = a a f() a f() ( D f() = a = a f() 3 f() a f() a a f() f() f(a) lim a f() (3) = a f() = a f() = e = lim f() = lim e = (3) = f() f() = f() = (pole) f() = a a = a lim f() = (3) a

13 a, ( a), ( a) 3, = a = a lim a f() a = a f() = e = x(x > ) = x(x < ) lim f() = lim e x x + x + = lim f() = lim x x + e x = (33) f() = ϕ() + c a + c ( a) + + c k ( a) k c m a f() f() = e = f() = e = + +! + 3! n! n + (34) = f(). ( ) 3

14 f(z) a Res[f, a] = c = f()d (35) πi a f() Res[f, a] = (36) a f() a f() k (k > ) Res[f, a] = lim a [( a)f()] (37) Res[f, a] = (k )! lim d k a d k [( a)k f()] (38) f() ( ) ()f() = e ( )( ) ( = ), () f() = ( + ) ( = ), (3) f() = e ( = ) () = f() Res[f, ] = lim( )f() = lim = e d () = f() Res[f, ] = lim d [ f()] = lim ( + ) = (3) ( a) (37) (38) (35) = [ e = + + ]! + + (n + )! n+ + = (n + )! n + = Res[f, ] = /6 f() e n f()d = πi Res[f, a k ] = Res[f, a ] + Res[f, a ] + + Res[f, a n ] (39) k= 4

15 a a m a () d : = 3, () ( + )( ) () =, Res[f, ] = lim ( + )f() = 3, () = i sin d : i = ( ) Res[f, ] = lim ( )f() = 3 d = πi(res[f, ] + Res[f, ]) = πi ( + )( ) Res[f, i] = lim d {( i) f()} = lim(sin + cos) = i e i i d sin d = πires[f, i] = πe ( ).3 (39) x f(x) 5

16 .3. f(x, Y ) π f(cosθ, sinθ) dθ cosθ, sinθ cosθ = (eiθ + e iθ ) = ( + ) sinθ = (eiθ e iθ ) = ( ) (4) i - sinθ cosθ π f(cosθ, sinθ)dθ = i f ( ( + ), ( )) i d, = eiθ (4) tan(θ/) = t I = π dθ 5 + 3sinθ I = π dθ (4) 5 + 3sinθ = e iθ (43) (4) theta π (4) (4) θ d dθ = ieiθ dθ = d (44) i sinθ = i (eiθ e iθ ) = ( ) (45) i I I = = π dθ 5 + 3sinθ = i 3 + i 3 d = i ( ) d 3( + i/3)( + 3i) (46) (46) = i/3 = 3i = i/3 Res[f(), i/3] = lim i/3 { ( + i 3 ) 3( + i/3)( + 3i) 6 } = 4i f() = 3 + i 3 (47)

17 3i i/3 θ π dθ 5 + 3cosθ = 3 d = πi + i 3 4i = π (48) 4 I = π dθ a + bcosθ (a > b > ) (4) I = π dθ a + bcosθ = i a + b ( + ) d = i d b + a + b iy i β α x i b + a + b = α, β α = a + a b, β = a a b b b I π dθ a + bcosθ = ib d ( α)( β) 7

18 = α, = β a > b > α, β α = a b ( ( ) ) b a < α < β < = α = α [ ] Res[f, α] = lim α ib ( α) ( α)( β) I = π dθ a + bcosθ = πi i a b = = ib α β = i a b π a b.3. f(x) x I = f(x)dx f(x) f() f() / I y a k a k R a R x f(x)dx = lim R f()d = πi n Res[f, a k ] (49) i=

19 I = dx x + x d = R R x + x + a dx + y + + d R R - 3i θ R x - + 3i R = Re iθ R I = + + d = d ( + 3i )( + + 3i ) = πires[f, 3i ] = π 3 Res[f, + 3i ] = I = πi = π 3i 3i 3 4 I = dx x + a (a > ) R R R R f()d = d R + a = R dx x + a + d + a (5) I (5) f() ±ia 9

20 y ia R R ia x ia d f()d = + a = d ( ia)( + ia) = πires[f, ia] = πi lim ( ia) ia ( + ia)( ia) (5) = π a (5) R I = Re iθ d π + a = Rie iθ R e iθ + a dθ = g(r, θ) R lim g(r, θ) = lim R R I = π a Rie iθ R e iθ + a = lim R π g(r, θ)dθ (5) ie iθ ie iθ = lim Reiθ R R = (53) ia I = πires[f, ia] = πi ia = π a (54) 5 I = e ikx x + a (a >, k ) e ik R f()d = + a d = e ikx x + a dx + e ik d (55) + a 5 k R

21 R = Re iθ e ik π + a d = e ikreiθ x + a Rieiθ dθ = π e ikrcosθ e krsinθ x + a Rie iθ dθ (56) π (56) e ikrcosθ e krsinθ x + a Rie iθ dθ e ikrcosθ e ikrcosθ = e krsinθ < θ < π sinθ k > k > R e krsinθ (55) k < I = πires[f, ia] = πi e ka ia = π a e ka (k > ) (57) ( ) π θ sinθ < R I = πires[f, ia] = πi eka ia = π a eka (k < ) (58) I = π a e k a Γ f() lim f() = Γ R f()e ikx d = (k > ) (59) lim R Γ /( + a e ik ) d = + a f()

22 y D Γ B R R θ A R x f() a k f() f()e imx dx = πi n Res[e imx f(), a k ] (6) k= 3 ( f(x) a lim f(x) = x a B A f(x)dx 6 B A f(x)dx = lim ϵ a ϵ A B f(x)dx + lim ϵ f(x)dx a+ϵ (6) ϵ, ϵ ϵ = ϵ = ϵ [ a ϵ B ] lim f(x)dx + f(x)dx ϵ A a+ϵ 6 (6)

23 [ a ϵ B ] B I = f(x)dx + f(x)dx = P f(x)dx (63) A a+ϵ P vp(value of principle) I = A dx (64) x x = [ϵ, ϵ ] ϵ ϵ ϵ ϵ ϵ, ϵ x = t I = lim ϵ,ϵ [ = lim ϵ I = [ ϵ log x ] ϵ = lim ϵ,ϵ log ϵ ϵ lim ϵ,ϵ ϵ x dx + x dx = + lim ϵ [ ϵ ϵ ϵ ] x dx [ log x x dx + t ( dt) = ] = lim ϵ,ϵ ϵ = ] ϵ x dx ϵ [ ϵ t dt x dx + ] ϵ x dx lim [log ϵ log ϵ ] ϵ,ϵ (65) (66) (66) ϵ = ϵ log(ϵ /ϵ ) = log k > kϵ = ϵ log ϵ /ϵ = log k K K (66) ϵ = ϵ (66) I 3

24 x dx (67) + 7 f(x)dx lim R R f(x)dx + lim R R f(x)dx dx x + = lim R R dx x + + lim R R dx x + = lim ( Arctan( R)) + lim Arctan(R) R R = π + π = π (68) [ Α ] [ Β ] -R ε a-ε a a -R R R a+ε [ a ϵ f(x) lim ϵ x a dx + a+ϵ ] f(x) x a dx (a ) (69) P [ a ϵ f(x) ] lim ϵ x a dx + f(x) x a dx = P a+ϵ f(x) dx (7) x a 7 Z Z xdx = lim R f(x)dx = lim 4 Z R R R xdx = lim R Z R R» x f(x)dx R R = lim R =

25 I = P f(x) dx (7) x a a ϵ R f() ( ) f() d = πi a a (7) f() a d = I + I + I = a ϵ R f(x) x a dx + R a+ϵ f(x) x a dx + I + I (73) R ϵ (69) I = a + ϵ e iθ (74) θ π I = π f(a + ϵe iθ ) ϵe iθ ϵ i ϵe iθ dθ = i π f(a + ϵe iθ )dθ (75) I = πif(a) (76) I R ;) (69) I = πif(a) + πi ( ) f() a (77) θ π π I = πif(a) πi ( ) f() a 5 (78)

26 ( k > I = P e ikx dx (79) x a k > 5 I = πif(a) + πi ( ) f() a (8) (8) I = πie ika (8) f() f(x) dx (a : ) (8) x a (principal value) P f(x) x a [ a ϵ dx = lim ϵ f(x) x a dx + a+ϵ ] f(x) x a dx (83) P f(x) x a = iπf(a) (84) lim ϵ + f(x) x a ± iϵ dx = P f(x) dx ıπf(a) (85) x a (

Tips KENZOU PC no problem 2 1 w = f(z) z 1 w w z w = (z z 0 ) b b w = log (z z 0 ) z = z 0 2π 2 z = z 0 w = z 1/2 z = re iθ θ (z = 0) 0 2π 0

Tips KENZOU PC no problem 2 1 w = f(z) z 1 w w z w = (z z 0 ) b b w = log (z z 0 ) z = z 0 2π 2 z = z 0 w = z 1/2 z = re iθ θ (z = 0) 0 2π 0 Tips KENZOU 28 7 6 P no problem 2 w = f(z) z w w z w = (z z ) b b w = log (z z ) z = z 2π 2 z = z w = z /2 z = re iθ θ (z = ) 2π 4π 2 θ θ 2π 4π z r re iθ re i2π = r re i4π = r w r re iθ/2 re iπ = r re