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

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

Advanced Engineering Mathematics
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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(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?
Transcribed Image Text:(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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