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PSN/CV-05001 Taylor series for $z^q(z-1)^p$

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Problem

Find the first three terms of the Taylor series of the function \(f(z)=z^p (z-1)^q\)
about  the point \(z_0=-2\). Assume the branch cuts for the
functions \(z^p\) and \((z-1)^q\) along the positive real axis. Write the
answer for the special case of \(p+q=1\)

Session Objective

In this problem solving session you will learn how to obtain Taylor expansion of a function with branch points.

Prerequisites

A good general understanding of functions with branch point and branch cuts is a must for this session.
If you are not sure of your understanding of  functions with branch cut and branch points, it is good
idea to make a beginning now. A good starting point is Chapter 3 [1]
Important: Several concepts and results about functions with branch points will be used here without any warning.

Skills required

  • How to define  function  \((z-a)^\lambda\) when branch cut is given
  • How to use definition and  compute values of the functions;
  • How to obtain Taylor Series by finding successive derivatives;

Quick Reminder

The Taylor series of a function \(f(z)\) in powers \((z-z_0)\) exists if
the given function is analytic  at \(z_0\) and is given by \begin{equation}  f(z) =\sum_{n=0}^\infty a_n (z-z_0)^n,
\end{equation} where \begin{equation}  a_n= \frac{1}{n!}\frac{d^n f(z)}{dz^n}\Big|_{z=z_0} \end{equation}


Exclude node summary : 

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Reference:
[1] A. K. Kapoor,  Complex Variables -- Principles and Problem Sessions,
World Scientific Singapore (2011).


 Document Type :: Problem Solving Session     Page-Id :: PSN/CV-05001   File-Id cv-psn-05001.pdf


 

Collection Root: 

Complex Variables :: Solutions and Errata

Complex Variables --- Principles and Problem Sessions  SOLUTIONS and ERRATA

Chapter-1 Complex Numbers

Chapter-2 :: Elementary Functions and Differentiation

Chapter 3 :: Functions with branch cut singularity

Chapter 4 :: Integration in Complex Plane

Chapter 5 :: Cauchy Integral Formula

Chapter 6 :: The Residue Theorem

Chapter 7 :: Contour Integration

Errata for Complex Variables ----Principles and Problem Sessions


This Collection has
(a) Solutions for Problem Sessions and
(b) Errata
for my 
book on complex variables.

PLEASE EXPAND THE TABLE OF CONTENT AND CLICK ON SECTION OF INTEREST


I take this opportunity to thank my friend and colleague, Prof. T. Amarnath, from School of Mathematics, University of Hyderabad for his kind words of appreciation and for recommending the book to National Board of Higher Mathematics for inclusion in scheme of distribution of mathematics books to Universities and Institutes in India.


 

 

Starting with this page the solutions to the problem sessions in the book

Complex Variables --- Principles and Prolems Sessions
published by World Scientific Singapore (2011)

are being made available for reference
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