homework108c4

Homework108c4 - /z around 0 and give its domain of convergence(3(This problem counts double Calculate using residue calculus the follow-ing

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HOMEWORK 4 FOR MA108C: COMPLEX ANALYSIS DUE DATE: 4PM, APRIL 28, 2009 (1) Let f j be holomorphic on U = D ( p,r ) \{ p } and suppose all f j have a removable singularity at p . Suppose f j f uniformly on compact subsets of U . Observe that f is holomorphic on U . What kind of singularity can f have at p ? The same question if all the f j have a pole at p , or all f j have an essential singularity at p . Is it possible for an infinite number of f j to have a pole, while another infinite number has an essential singularity? (2) (a) Calculate the first four terms (i.e. lowest four terms) of the Laurent series expansion of f ( z ) = z/ sinh( z ) 2 in the two annuli around 0 with smallest radii, such that the convergence of the Laurent series cannot be extended to larger annuli; i.e. find 0 < r 1 < r 2 , and two different Laurent series which converge to f on A (0 , 0 ,r 1 ), respectively A (0 ,r 1 ,r 2 ). (b) Calculate the Laurent series expansion of sin(1
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Unformatted text preview: /z ) around 0, and give its domain of convergence. (3) (This problem counts double.) Calculate using residue calculus the follow-ing integrals (a) H γ ( e z-1)(1 + z ) /z ( z-2 i ) 2 ( z + 3) cos( πz/ 2) dz for the contour γ as depicted below (on the back side); (b) R ∞ 1 / (1 + x 3 ) dx ; (c) R ∞-∞ x/ sinh( x ) dx (recall sinh( x ) = ( e x-e-x ) / 2); (d) R π-π 1 / ( a + b cos( θ )) dθ for a > | b | and a,b ∈ R (Hint: Unit circle); Finally, calculate the sum ∑ n ∈ Z 1 / ( u + n ) 2 ( u 6∈ Z ) by integrating π cot( πz ) / ( u + z ) 2 over the circle ∂D (0 ,N + 1 / 2) for integer N . 1 2 HOMEWORK 4 FOR MA108C: COMPLEX ANALYSIS DUE DATE: 4PM, APRIL 28, 2009 Figure 1. The contour used in problem 3a...
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Homework108c4 - /z around 0 and give its domain of convergence(3(This problem counts double Calculate using residue calculus the follow-ing

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