(d) Find the Laurent series of the function f(z): about z = expression for the exponential function. Where does the series converge? = 0 by using the well-known
(d) Find the Laurent series of the function f(z): about z = expression for the exponential function. Where does the series converge? = 0 by using the well-known
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
question a and d
![(a) Find the derivative of the power series
f(z) = −3+
=
(b) Assume a function f is analytic at z
derivatives at the origin are given by f(n) (0)
2 = 0.
iloi
∞
2πί
n=]
3n
0, where f(0)
inn!
n³
(c) Assume a function f is analytic in some neighbourhood of z = √2. Assume further that
1
f(z)
(z - √2)n+
= 5. Assume further that its
for n 1. Find its Taylor series about
-dz =
(n + 5) / /
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
n
[ f(²)(z − √2)" dz
-
e-
for all n E N.
(d) Find the Laurent series of the function f(z)
expression for the exponential function. Where does the series converge?
25
-3z
about z = 0 by using the well-known](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a7d82e4-2bf5-4850-ba06-9621bb0bd1d8%2F9c68d08b-abde-4d34-b257-bed2731e072e%2Fw3mdrc9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Find the derivative of the power series
f(z) = −3+
=
(b) Assume a function f is analytic at z
derivatives at the origin are given by f(n) (0)
2 = 0.
iloi
∞
2πί
n=]
3n
0, where f(0)
inn!
n³
(c) Assume a function f is analytic in some neighbourhood of z = √2. Assume further that
1
f(z)
(z - √2)n+
= 5. Assume further that its
for n 1. Find its Taylor series about
-dz =
(n + 5) / /
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
n
[ f(²)(z − √2)" dz
-
e-
for all n E N.
(d) Find the Laurent series of the function f(z)
expression for the exponential function. Where does the series converge?
25
-3z
about z = 0 by using the well-known
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