Problems-2

Problems-2 - ∞-∞ 1 ( x 3-1) dx, p.v. Z ∞-∞ sin x x...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Practice problems for the Midterm 2. 1. Find singularities of the function and describe their type: f ( z ) = sin( 1 z ) ( πz ) 2 - 1 , 1 cos( z ) - 1 - z 2 / 2 f ( z ) = f ( z ) = sin( z ) z ( z + π )( z - 1) , z 6 = 0 , 1 , - π 1 , z = 0 - 1 π ( π +1) , z = - π 2. Find Laurent series for z z 4 + 1 , about z = 0 , 1 ( z - 1)( z 2 - 1) , about z = 1 , sin( z 2 ) z 3 , about z = 0 , sin( πz 2 ) 1 + z , about z = - 1 , 3. Find residues of functions at all their singular points ( z + 1) cos( 1 z ) , sin 2 ( z 2 ) 1 - cos z , z + 1 ( z - 1) 3 z ze 1 z , ( z + z 2 ) e 1 z , e z + 1 z 4. Compute the following improper integrals using the residues Z -∞ cos x x 2 + 1 dx, Z -∞ x sin x ( x 2 + 1) 2 dx, p.v. Z
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ∞-∞ 1 ( x 3-1) dx, p.v. Z ∞-∞ sin x x ( x 2 + 1 ) dx 5. Let C be the counter clock wise oriented circle around the origin of radius 2. For each function f indicate if it is true or false that R C f ( z ) dz = 0. f ( z ) = 1 z ( z-1) , f ( z ) = sin z, f ( z ) = tan(10 z ) , f ( z ) = 1 z 2 + 5 6. Find the decomposition into simple fractions for 1 z 2 ( z-1) 2 , z z 3 ( z-2) 1...
View Full Document

This note was uploaded on 05/03/2010 for the course MATH 185 taught by Professor Lim during the Summer '07 term at Berkeley.

Ask a homework question - tutors are online