Understanding Taylor Series in Complex Analysis

Sep 15, 2024

Lecture Notes: Taylor Series in Complex Analysis

Overview

  • Focus on Taylor Series and its application in complex analysis.
  • Introduction to two types of series: Taylor and Laurent Series.

Previous Lectures Recap

  1. Cauchy-Riemann Conditions: Identifying whether a function is analytic.
  2. Finding Imaginary/Real Parts: How to derive one part if the other is given for analytic functions.
  3. Milne-Thomson Method: Constructing a function given its real part.
  4. Linear Transformation: Explaining transformations in analytic functions.

Taylor Series Definition

  • A function is analytic inside a circle centered at z=a, allowing for expansion in a Taylor Series.
  • Important: The point must lie within the analytic function; otherwise, the series cannot be expressed.

Methods to Solve Taylor Series Problems

Method 1: Differentiation Method

  1. Differentiate the function multiple times to find f'(a) and higher derivatives.
  2. Use the Taylor series formula with these derivatives.
  3. Example Calculation:
    • Set z = 1 for specific calculations.
    • Adjust constants as necessary for simplification.

Method 2: Binomial Expansion

  • Simplifies the process for certain problems.
  • Use the Binomial theorem for expansions.
  • If given z = t + 1, reformulate to accommodate binomial expansion.

Example Problems

Problem 1

  • Task: Find first four terms of the Taylor series expansion about z=2.
  • Identify Singularities: Points where function is not analytic (e.g., z=3 and z=4).
  • Circle Center: z=2 with radius ensuring singularities remain outside the circle of convergence.

Problem 2

  • Similar to Problem 1, ensuring region of convergence excludes singularities (1 and 3).
  • The radius for the circle is chosen to ensure points lie outside.
  • Reiterate: Use both methods to derive series, but note that differentiation is more complicated.

Conclusion

  • Importance of ensuring function's analyticity within the defined region for valid Taylor series expansion.
  • Upcoming Topic: Laurent Series - expansion in specified regions.
  • Reminder to check previous videos for a more in-depth understanding of the discussed concepts.

Engagement

  • Encourage students to comment, share, and like videos for continued support.