The hyperbolic sine is the function defined by e? – e-z sinh(z):= for all z E C. (a) After checking that sinh(z) = -i sin(iz), calculate the Taylor series of sinh(z) centered at zo = 0, specifying its domain of convergence. (b) Determine the domain of holomorphy of the function + 2 cos(z) – 3 23 sinh(z) f(2) = Then, show that zo = 0 is a removable singularity for f, and determine the analytic extension g(z) of f(z) in a suitable neighborhood of zo = 0. (c) Compute g'(0).
The hyperbolic sine is the function defined by e? – e-z sinh(z):= for all z E C. (a) After checking that sinh(z) = -i sin(iz), calculate the Taylor series of sinh(z) centered at zo = 0, specifying its domain of convergence. (b) Determine the domain of holomorphy of the function + 2 cos(z) – 3 23 sinh(z) f(2) = Then, show that zo = 0 is a removable singularity for f, and determine the analytic extension g(z) of f(z) in a suitable neighborhood of zo = 0. (c) Compute g'(0).
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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Transcribed Image Text:The hyperbolic sine is the function defined by
e – e-z
sinh(z) :=
2
for all z E C.
(a) After checking that sinh(z) = -i sin(iz), calculate the Taylor series of sinh(z) centered
at zo = 0, specifying its domain of convergence.
(b) Determine the domain of holomorphy of the function
+ 2 cos(z) – 3
23 sinh(z)
f(2) =
Then, show that zo = 0 is a removable singularity for f, and determine the analytic
extension g(z) of f(z) in a suitable neighborhood of zo = 0.
(c) Compute g'(0).
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