chap6

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

This preview shows pages 1–10. Sign up to view the full content.

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 zε < - < Singular points C 1 1 Z 0 L ( 1 1 ) 1 f ÿ¼ ª*

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 1 z-2 1 ) 1 16 1 1 1 but 4 4 2 ( 2) ( ( 1 1 4 2 2( 1 ( 2 ( 1) 4 ( 0 2 2 1 2 0 1 2 16 2 ( . z z z- z . z z- n - n- n n dz b i c π - = + - - = - - - = < < + = =- = 8 π i = - 2 Re 1 has singular points at 0 2 4 ( it has Laurent series representation in 0 2 2 1 based on (2) 2 4 4 ( ( z s z , z z z dz πi c z z z z = = = - < < = - - 2 1 4 ( dz C: z- c z z = - Ex4. 0 2 4
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 =
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 dz Z Z f(z) ) π i f(z dz π Z Z ) f(z c Z-Z dz ) but f(z dz c Z Z ) f(z f(z)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/20/2008 for the course MATH 61 taught by Professor Enderson during the Fall '08 term at UCLA.

### Page1 / 45

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

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online