(a) Find the derivative of the power series 1 f(z) = −3+ n=1 #z”. n
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:Problem 3
(a) Find the derivative of the power series
∞ 1
f(x) = −3+ Σ
n=1
(b) Assume a function f is analytic at z = 0, where f(0) 5. Assume further that its
derivatives at the origin are given by f(n) (0)
inn!
=
for n ≥ 1. Find its Taylor series about
n³
2 = 0.
(c) Assume a function f is analytic in some neighbourhood of z = √2. Assume further that
1
f(z)
(2-√√2)n+1
i So
2πi
-dz
=
(n + 5) ³/
for n ≥ 0, where C is a positively oriented circle of radius e centred at z = √2. Find the
Taylor series about z = √2 and evaluate the integrals
[1(²)(z-√2)" dz
-
for all n E N.
(d) Find the Laurent series of the function f(2)=
expression for the exponential function. Where does the series converge?
about z = 0 by using the well-known
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