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
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.

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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