I ( ) ( ) (1) C z = a ρ. f(z) dz = C = = (z a) n dz C n= p 2π (ρe iθ ) n ρie iθ dθ 0 n= p { 2πiA 1 n = 1 0 n 1 (2) C f(z) n.. n f(z)dz = 2πi Re

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1 I ( ). ( ) () a ρ. f() d ( a) n d n p π (ρe iθ ) n ρie iθ dθ n p { πia n n () f() n.. n f()d πi es f( k ) k n n. f()d n k k f()d. n f()d πi esf( k ). k

2 I ( ). ( ) () f() p g() f() g()( ) p. f(). f() A p ( ) p + A p+ ( ) p+ + f()( ) p A p + A p+ ( ) + + A p+k ( ) k + A p ( ) p f(), A p+ A d d [( ) p f()], d p (p )! d [( ) p f()]. p () f(). Q() (). ( ) dq() d Q() ()( ) es f( ) ( )f() (.) P () ( ) ()( ) (.) P ( ) ( ). (.3) [ ()( ) + ()] es f( ) P ( ) Q ( ). (3) (i) f() 3 + 5,. ( ) 3 k f() { 5 k k

3 I ( ) 3 ( )k f() { 6 k 3 k 3 3. (ii) f() cos 3 3. (iii) f() sin es f() 5 es f()!! 6 d d [( )3 f()] ( 3 ) + 5 d d. k f() es f()!! d d [3 f()] { k 3 k 3 d { 3 (! d 3! + 4! 4 + d (! d! + ) 4! 4 + ) ( +! + )} nπ. ( k( nπ) k nπ nπ)k f() nπ cos { ( ) n k k. es f(nπ) ( ) n (iv) f() e ( ) + e n i + n π. { ( ) k i π} + n f() (/+n)π (/+n)π { k i (/+n)π { ( ) } k i + n e π + e ( + n ) π } k e { e + ( ) i + n } k ( + )e π e

4 I ( ) 4 k. k e (/+n)π e (/+n)π { i e ( ) + n π } e i ( +n )π ( ) + n π i( ) n ( + n ) π i( ) n. ( + n ) π i( ) n es f((/ + n)π)

5 I ( ) 5.3 ( ) () f() M α M α M. I f() d α < I f() d f() d θb θ a M α dθ M α+ (θ b θ a ). I. Im θb O θa e.: () f() > N N f() < ε ε. I e ia f() d

6 I ( ) 6 (.4) I e ia f() d e ia f() d. e iθ > N f() ε e ia e iaeiθ e ia(cos θ+i sin θ) a sin θ e d dθ π π/ I < ε e a sin θ dθ ε e a sin θ dθ. θ π/ Jordan π/ e a sin θ dθ π/ π θ sin θ e a π θ dθ π a ( e a ) < π a. I < επ a. ε I (3) a < π arg I e ia f() d. () I < ε e a sin θ dθ ε e a sin θ dθ. π π/

7 I ( ) 7 Im O θ e.:. π/ θ a < π/ e a sin θ dθ π/ π θ sin θ e a π θ dθ π a ( e a ) <. (). π a.

8 I ( ) 8.4 () ( ) () e iθ d ie iθ dθ i dθ. I f ( + cos θ +, sin θ i, ) d i i ( + f, ) d i i.. () (i) e iθ π dθ a + b cos θ i d i{a + b( + )/} d i{a + b( + )/} d b + a + b.. ± a ± a b b +. [ ] es b + a + b + a b π dθ a + b cos θ π a b (ii) (i) e iθ. cos nθ (e inθ + e inθ )/ ( n + n )/ π cos 3θ a cos θ + a dθ ( ) /4 d a( + )/ + a i ( 6 + ) d 4ai 5 ( a)( a )

9 I ( ) 9 > a 5 ± a ( ). f() es f(± a) (a3 + ) 8ia 3 (a ), es f() ( a + + /a ) 4ai ( I πi (a3 + ) 8ia 3 (a ) + a + + /a ) 4ai π( a + a ). a

10 I ( ).5 () ( ) ().3 f() d f() d + f() d. e iθ ( θ π) π f() d i f(e iθ ) dθ π f() d f(e iθ ) dθ. f() O( α )(α < ) f(e iθ ) M M. f() d M π.. I f() d Im Zk O θ e.3:

11 I ( ) () (ii) f() O( α )(α < )..3 I + a d f() ia ( ). I πies f(ia) π a. (ii) f() O( α )(α < ) a d a d. 4 I 4 + a 4 d.3. f() ae iπ/4 ae i3π/4 ( ). es f(ae iπ/4 ) e i3π/4 4a 3 i, I πi es f(ae 3iπ/4 ) e iπ/4 4a 3 i, [ e i3π/4 4a 3 i ] + e iπ/4 4a 3 i π 4a 3.

12 I ( ).6 (3) ( ) ().3 (). () (i) f() e.. π/4. OA,, y x BO. e d d + e d + d OA e BO e. OA e d e x dx π ( ), e d π/4 π/4 π/4 e e iθ e iθ dθ e e iθ e iθ dθ e dθ π 4 e ( ), BO e iπ/4, d e iπ/4 d BO e d e e iπ/ e iπ/4 d + i e i d + i (cos i sin ) d + i (cos i sin ) d, ( ), + i (cos i sin ) d π sin(x ) dx π. (ii) e iax dx (b > ) x ib

13 I ( ) 3 a >.3. e ia ib d e ia ib d + e ia ib d πie ab ( ) e iθ, d e iθ dθ e ia π ib d e a sin θ+ia cos θ ie iθ dθ. e iθ ib e ia ib d < π π e iθ ib e a sin θ dθ e a sin θ dθ π/ π/ e a sin θ dθ e (/π)aθ dθ π a ( e a ) ( ). e iax x ib dx πie ab. a < e iax dx. x ib

14 I ( ) 4.7 ( ) Im O Z γ ε e.4: (). () f() ( ) (es f( )) A. ε γ γ f() d (3) (), () f() d γ π γ A π A d εe iθ εeiθ dθ iπes f( ) A π εe iθ εeiθ dθ A dθ π A dθ iπes f( ) m P f(x)dx πies f( ) + πi es f( k ) k (4) (i) a..4 e im a e im a ε a d + e im a d + a+ε e im a d + γ e im d. a

15 I ( ) 5 e im d a γ e im a ε d πies a γ πie ima e im e im d πieima a e im a d + a+ε ε, P a d m sin θ e a d m sin θ e + a d m sin θ e + a π [ e im a ] a e im d πieima a e im a d πieima π(i cos ma sin ma) cos mx P dx π sin ma x a (ii)..4 e i ε d + e i d + e i +ε γ d + e i d. e i d e i d sin θ e d sin θ e sin θ e d π

16 I ( ) 6 γ e i d πies πi [ ] e i γ e i d πi ε e i d + +ε ε, e i d πi P e i d πi (iii) (ii) ε e i d + +ε e i d πi ε e i +ε d e iζ ζ dζ e iζ dζ +ε ζ +ε e i e i d +ε i sin d πi +ε sin ε, sin d π d π

4 R f(x)dx = f(z) f(z) R f(z) = lim R f(x) p(x) q(x) f(x) = p(x) q(x) = [ q(x) [ p(x) + p(x) [ q(x) dx =πi Res(z ) + Res(z )+ + Res(z n ) Res(z k ) k

f(x) f(z) z = x + i f(z). x f(x) + R f(x)dx = lim f(x)dx. R + f(x)dx = = lim R f(x)dx + f(x)dx f(x)dx + lim R R f(x)dx Im z R Re z.: +R. R f(z) = R f(x)dx + f(z) 3 4 R f(x)dx = f(z) f(z) R f(z) = lim R