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chap6 - Chap 6 Residues and Poles Cauchy-Goursat Theorem...

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1 Chap 6 Residues and Poles Cauchy-Goursat Theorem: = c f dz 0 if f analytic. What if f is not analytic at finite number of points interior to C Residues. 53. Residues z 0 is called a singular point of a function f if f fails to be analytic at z 0 but is analytic at some point in every neighborhood of z 0 . A singular point z 0 is said to be isolated if, in addition, there is a deleted neighborhood of z 0 throughout which f is analytic. 0 0 z < - < Singular points C 1 1 Z 0 L ( 1 1 ) 1 f ÿ¼ ª*
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2 Ex1. 2 2 1 has isolated singnlar points 0 1 z z z , i (z ) + = + Ex2. The origin is a singular point of Log z , but is not isolated Ex3. 1 sin( ) 1 singular points 0 and 1 2 . z z z n , , ... n π = = = not isolated isolated When z 0 is an isolated singular point of a function f , there is a R 2 such that f is analytic in 0 2 0 z z R < - < 0
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3 Consequently, f ( z ) is represented by a Laurent series 1 2 0 2 0 0 0 0 2 ( ) ( ) ..... ....... (1) ( ) ( ) 0 0 ( ) 1 where 2 ( n n n b b b n f z a z z z z z z z z n z z R f z dz b n i z π = - + + + + + - - - = < - < = - 0 ( 1, 2, ... ) c 1 ) n n z = - + and C is positively oriented simple closed contour 0 0 2 around and lying in 0 z z z R < - < When n =1, 1 2 ( ) (2) πi b f z dz c = The complex number b 1 , which is the coefficient of in expansion (1) , is called the residue of f at the isolated singular point z 0 . 0 1 z z - 0 Re ( ) z z s f z = A powerful tool for evaluating certain integrals. 0 2 0 z z R < - < R.O.C.
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4 1 z-2 1 ) 1 16 1 1 1 but 4 4 2 ( 2) ( 2) ( 2) 1 1 4 2 2( 2) 1 ( 2 ( 1) 4 ( 2) 0 2 2 1 2 0 1 2 16 2 ( 2) . z z z- z . z z- n - n- z- z- n n dz b i c z z- π - = + - - = - - - = < < + = =- = 8 π i = - 2 Re 1 has singular points at 0 2 4 ( 2) it has Laurent series representation in 0 2 2 1 based on (2) 2 4 4 ( 2) ( 2) z s z , z z z z- dz πi c z z z z = = = - < < = - - 2 1 4 ( 2) dz C: z- c z z = - Ex4. 0 2 4
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5 1 ) 2 3 1 1! 2! 3! 1 1 1 1 1 1 2 1 0 2 2! 4 3! 6 1! 0 1 exp ( 0 2 analytic on and within z z z z z z e ...... z e ..... z z z b dz c z f C f = + + + + < = + + + + < < = = The reverse is not necessarily true. 0 dz c = 2 2 ( ) 1 show exp 0 where 1 1 is analytic everywhere except at the origin dz c z C: z z = = Ex5 . 0
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6 More on Cauchy Integral Formula (1) Simply-Connected and Multiply-Connected Simply Connected Multiply Connected ( ) ( ) 0 1 k n f z dz f z dz c c k + = = C 1 C 2 C ( ) 0 f z dz c = C
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7 C0 C ........ same. dz c Z π θ i e i ρ π ρe c Z dz π i c z dz π i π ρ e iρρ c z dz ce ρe Z π i c z dz to show 0 2 0 2 0 2 2 2 0 0 2 0 Z at except everywhere analytic is Z 1 and 2 2 0 0 sin , 0 C circle a stmct con 2 = = = = = = = = = = More on Cauchy Integral Formula (2) Simply-Connected and Multiply-Connected
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8 ( ) 0 ( ) 0 0 0 ( ) 0 0 ( ) ( ) ) ( ) ) 0) ( ) ( ! 0 or ( ! 0 Replace by - ( ) ( ! 0 n n n n n n z z g g z z z R n n f f z z z n n z z z , f f z z z n n f = < = + = = = - = More on Cauchy Integral Formula (3) Simply-Connected and Multiply-Connected f(z) c f(s) ds π i Z S = - 2 1 ) π i f(z c Z-Z ) dz f(z c Z-Z f(z) dz 0 2 0 0 0 = = π i c Z-Z z z d c Z-Z dz c Z dz 2 0 ) 0 ( 0 = - = = Then connection to Taylor Series……. 1 1 0 0 0 0 ( ) 1 0 0 ( ) 1 0 0 ( ) (0) 2 ! (0) ( ) ! ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) 2 wher N n N c n N n n N n N N n n N n N n f s ds s f i n f z z n f s ds f s ds z z c c s z s z s f s ds z z c s z s f s ds f z ρ c s z πi π - + = - = - = = + - - = + - = = + - 0 ( ) ( ) e 2 ( ) N N N z f s ds z ρ c πi s z s = - ) π i f(z c Z-Z ) dz f(z 0 2 0 0 =
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9 Why (chap4) = < - - = - - < - - - - = - = - = - - = π πρ ρ dz c Z Z ) f(z f(z) dz c Z Z ) f(z f(z) c dz Z Z ) f(z f(z) btu dz c Z Z ) f(z f(z) ) π i f(z - c
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