questionbcd (a) Find the derivative of the power series∞ 1f(z) = −3+2πίn=1(b) Assume a function f is analytic at z = 0, where f(0)derivatives at the origin are given by f(n) (0)in n!n³z = 0.i Jo3n(c) Assume a function f is analytic in some neighbourhood of z = √2. Assume further that1f(z)(z - √2)n+1= 5. Assume further that itsfor n 1. Find its Taylor series about-dz = (n + 5) ³/for n ≥ 0, where C is a positively oriented circle of radius e centred at z = √2. Find theTaylor series about z = √2 and evaluate the integrals[ f(z) (z − √2)" dz=for all n E N.-3z(d) Find the Laurent series of the function f(z)about z = 0 by using the well-knownz5expression for the exponential function. Where does the series converge?

Answered: (a) Find the derivative of the power…

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(a) Find the derivative of the power series 1 f(z) = −3+ In ² n=1

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter2: Graphical And Tabular Analysis

Section2.1: Tables And Trends

Problem 1TU: If a coffee filter is dropped, its velocity after t seconds is given by v(t)=4(10.0003t) feet per...

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Question

questionbcd

(a) Find the derivative of the power series
∞ 1
f(z) = −3+
2πί
n=1
(b) Assume a function f is analytic at z = 0, where f(0)
derivatives at the origin are given by f(n) (0)
in n!
n³
z = 0.
i Jo
3n
(c) Assume a function f is analytic in some neighbourhood of z = √2. Assume further that
1
f(z)
(z - √2)n+1
= 5. Assume further that its
for n 1. Find its Taylor series about
-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
[ f(z) (z − √2)" dz
=
for all n E N.
-3z
(d) Find the Laurent series of the function f(z)
about z = 0 by using the well-known
z5
expression for the exponential function. Where does the series converge?

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