M461practicesol2

# M461practicesol2 - MAT 461 Test 2 Practice Posted November...

This preview shows pages 1–3. Sign up to view the full content.

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.

Unformatted text preview: MAT 461 - Test 2 Practice Posted November 13 1. (a) Let f ( z ) = 1 z ( z- 1) . Give the two Laurent Series centered at z = 0, one valid < | z | < 1, one valid 1 < | z | < ∞ . (b) Give the Laurent series centered at z = π 2 for f ( z ) = sin z ( z- π 2 ) 3 . What is the residue of f ( z ) and z = π 2 ? 2. Let f ( z ) be entire. Assume that | f ( z ) | > m for all complex numbers z , where m is a positive constant. Prove that f ( z ) is constant. Let C be the circle | z | = 2 , positively oriented. 3. Compute the integral integraldisplay C 1 z 2 ( z − 3 i ) dz 4. Compute the integral integraldisplay C ( z − 1) 3 e 1 z- 1 dz 5. Compute the integral integraldisplay C e 1 z cos( 1 z ) dz 6. Define log z with the branch cut at − π < θ < π . Compute the integral integraldisplay C sin(log( z + 3)) ( z − 4 i ) 2 dz Solutions 1. (a) 1 z ( z − 1) = 1 z − 1 1 − z = 1 z ∞ summationdisplay n =0 − z n this sum works for | z | < 1 = − 1 z − 1 − z − z 2 − · · · =...
View Full Document

## This note was uploaded on 05/11/2008 for the course MAT 461 taught by Professor Ibrahim during the Fall '07 term at ASU.

### Page1 / 4

M461practicesol2 - MAT 461 Test 2 Practice Posted November...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online