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6. This problem concerns the evaluation of sums using contour integration. (a) The function g(z) = π cot Tz has a period of unity and poles at z = j, where j is any integer with -

6. This problem concerns the evaluation of sums using contour integration. (a) The function g(z) = π cot Tz has a period of unity and poles at z = j, where j is any integer with -

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 6.
6. This problem concerns the evaluation of sums using contour integration. (a) The function g(z)= π cot az has a period of unity and poles at z = j, where j is any integer with - <j<∞o. What are the residues of g(z) at its poles? (b) Now suppose that we want to evaluate S = f(z) has only isolated singularities at the points zx and |zf(z)| → 0 for large 2. The latter requirement is sufficient to ensure the sum is abso- lutely convergent. Consider also the contour integral [ f(n). 8118 I = - =$.$( where is circle centered at z = 0 with radius R = m +, where m is a large positive ineger. This formulation for C avoids the singularities of cot z. Show that I 0 as m →∞. (c) Using the residue theorem, show also that Q Search f(2) π cot z dz, 1–2πί Σ f(n) + Σresidues of f(2) π cot nz at zu (d) Use this result to evaluate as moo. Thus we can conclude S = -residues of f(2) π cot z at zk. ∞ S= Σ n=18 1 n²+ a²¹ where a is a complex constant, with a² a non-positive integer. r " Q
Transcribed Image Text:6. This problem concerns the evaluation of sums using contour integration. (a) The function g(z)= π cot az has a period of unity and poles at z = j, where j is any integer with - <j<∞o. What are the residues of g(z) at its poles? (b) Now suppose that we want to evaluate S = f(z) has only isolated singularities at the points zx and |zf(z)| → 0 for large 2. The latter requirement is sufficient to ensure the sum is abso- lutely convergent. Consider also the contour integral [ f(n). 8118 I = - =$.$( where is circle centered at z = 0 with radius R = m +, where m is a large positive ineger. This formulation for C avoids the singularities of cot z. Show that I 0 as m →∞. (c) Using the residue theorem, show also that Q Search f(2) π cot z dz, 1–2πί Σ f(n) + Σresidues of f(2) π cot nz at zu (d) Use this result to evaluate as moo. Thus we can conclude S = -residues of f(2) π cot z at zk. ∞ S= Σ n=18 1 n²+ a²¹ where a is a complex constant, with a² a non-positive integer. r " Q
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