Problems-2

# Problems-2 - ∞-∞ 1 x 3-1 dx p.v Z ∞-∞ sin x x x 2 1...

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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.
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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...
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