📐Complex Analysis Unit 6 – Analytic Function Series Representations

Key Concepts and Definitions

• Analytic functions are complex-valued functions that are differentiable in a neighborhood of every point in their domain
• Power series are infinite series of the form $\sum_{n=0}^{\infty} a_n (z-z_0)^n$ where $a_n$ are complex coefficients and $z_0$ is the center of the series
• Taylor series are power series expansions of analytic functions around a point $z_0$ with coefficients determined by the function's derivatives at $z_0$
• Laurent series are power series expansions that allow for negative powers of $(z-z_0)$ and are used to represent functions with isolated singularities
• Radius of convergence is the largest radius $R$ such that a power series converges for all points $z$ satisfying $|z-z_0| < R$
• Cauchy's Integral Formula expresses the value of an analytic function at a point in terms of a contour integral of the function over a closed curve surrounding the point
• Residue Theorem relates the residues of a meromorphic function to the contour integral of the function over a closed curve

Power Series in the Complex Plane

• Power series in the complex plane are a way to represent analytic functions as infinite sums of powers of $(z-z_0)$
• The coefficients $a_n$ of a power series can be complex numbers, allowing for a wider range of functions to be represented compared to real power series
• Power series can be used to define new functions, such as the exponential function $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$ and the trigonometric functions $\sin(z)$ and $\cos(z)$
• The convergence of a power series depends on the values of $z$ and the behavior of the coefficients $a_n$ as $n$ approaches infinity
  • A power series may converge for all $z \in \mathbb{C}$ (entire function), for $|z-z_0| < R$ (disk of convergence), or only at $z_0$
• Operations such as addition, multiplication, and differentiation can be performed term-by-term on power series within their disk of convergence
• The Ratio Test and Root Test are often used to determine the radius of convergence of a power series

Taylor Series for Analytic Functions

• Taylor series are a special case of power series where the coefficients are determined by the derivatives of the function at the center point $z_0$
• For an analytic function $f(z)$, the Taylor series expansion around $z_0$ is given by $f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z-z_0)^n$
  • The coefficients $\frac{f^{(n)}(z_0)}{n!}$ are called the Taylor coefficients of $f$ at $z_0$
• If a function is analytic in a disk centered at $z_0$, its Taylor series converges to the function within that disk
• Taylor series can be used to approximate functions, solve differential equations, and study the local behavior of functions near a point
• Examples of Taylor series include the Maclaurin series (Taylor series centered at $z_0=0$) for $e^z$, $\sin(z)$, and $\cos(z)$
• The error in a Taylor polynomial approximation can be estimated using the Lagrange Remainder Theorem or the Cauchy Remainder Theorem

Laurent Series and Isolated Singularities

• Laurent series are an extension of Taylor series that allow for negative powers of $(z-z_0)$ in the series expansion
• Laurent series are used to represent functions with isolated singularities, which are points where the function is not analytic but is analytic in a punctured disk around the point
• The Laurent series of a function $f(z)$ around a point $z_0$ is given by $f(z) = \sum_{n=-\infty}^{\infty} a_n (z-z_0)^n$
  • The coefficients $a_n$ for $n < 0$ are called the principal part of the Laurent series
  • The coefficients $a_n$ for $n \geq 0$ form the analytic part of the Laurent series
• Isolated singularities can be classified based on the principal part of the Laurent series:
  • Removable singularity: Principal part is zero ($a_n = 0$ for all $n < 0$)
  • Pole of order $m$: Principal part has a finite number of non-zero terms, with the lowest power being $-m$
  • Essential singularity: Principal part has infinitely many non-zero terms
• The Residue Theorem relates the residue (coefficient $a_{-1}$) of a Laurent series to contour integrals of the function around the singularity

Convergence and Radius of Convergence

• The convergence of a power series or Laurent series determines the values of $z$ for which the series represents the function
• The radius of convergence $R$ is the largest radius such that the series converges for all points $z$ satisfying $|z-z_0| < R$
  • Within the disk of convergence, the series converges to the function
  • On the boundary of the disk ($|z-z_0| = R$), the series may converge conditionally, diverge, or converge to a different value than the function
  • Outside the disk of convergence, the series diverges
• The Ratio Test is often used to find the radius of convergence:
  • $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L$
  • If $L < 1$, the series converges for $|z-z_0| < \frac{1}{L}$
  • If $L > 1$, the series diverges for $|z-z_0| > \frac{1}{L}$
  • If $L = 1$, the test is inconclusive, and other methods (such as the Root Test) must be used
• The radius of convergence can also be determined by finding the distance from the center $z_0$ to the nearest singularity of the function
• Cauchy-Hadamard Theorem provides another method for finding the radius of convergence using the limit superior of the coefficients

Applications in Complex Analysis

• Power series and Laurent series are essential tools in complex analysis for representing and studying the behavior of analytic functions
• Cauchy's Integral Formula can be derived using power series expansions and allows for the computation of higher-order derivatives of analytic functions
• The Residue Theorem is a powerful tool for evaluating contour integrals of meromorphic functions and has applications in physics and engineering
  • Residues can be used to find the sum of series, evaluate improper integrals, and solve differential equations
• Conformal mapping, which preserves angles and shapes of infinitesimal figures, can be studied using power series expansions
  • The Riemann Mapping Theorem guarantees the existence of a conformal map between any simply connected domain (other than $\mathbb{C}$ itself) and the unit disk
• Power series and Laurent series are used in the study of entire functions, meromorphic functions, and the distribution of zeros and singularities
• The Weierstrass Factorization Theorem states that any entire function can be represented as a product of factors involving its zeros
• The study of analytic continuation, which extends the domain of an analytic function, relies on the use of power series expansions

Common Pitfalls and Misconceptions

• Confusing the concepts of analyticity and differentiability: A function may be differentiable at a point without being analytic in a neighborhood of that point
• Misunderstanding the role of the center $z_0$ in power series and Laurent series expansions
  • The choice of $z_0$ affects the coefficients and the region of convergence
• Incorrectly applying the Ratio Test or Root Test for convergence
  • The limit must be taken as $n \to \infty$, not as $z \to \infty$
  • The tests provide information about the radius of convergence, not the convergence at a specific point
• Misinterpreting the meaning of the principal part in a Laurent series
  • The principal part corresponds to the singular behavior of the function near the isolated singularity
• Forgetting to consider the behavior of the series on the boundary of the disk of convergence
  • Convergence on the boundary requires additional analysis and may differ from the behavior within the disk
• Misapplying the Residue Theorem by not considering the multiplicity of poles or the presence of essential singularities
• Confusing the concepts of uniform convergence and pointwise convergence for sequences and series of functions

Practice Problems and Examples

1. Find the Taylor series expansion of $f(z) = \frac{1}{1-z}$ around $z_0 = 0$ and determine its radius of convergence.
2. Determine the Laurent series expansion of $f(z) = \frac{1}{z(z-1)}$ around $z_0 = 0$ and classify the singularities at $z = 0$ and $z = 1$.
3. Use the Residue Theorem to evaluate the contour integral $\oint_C \frac{\sin(z)}{z^2} dz$, where $C$ is the circle $|z| = 2$ oriented counterclockwise.
4. Find the Maclaurin series for $\cos(z)$ and use it to approximate $\cos(0.1)$ with an error less than $10^{-6}$.
5. Prove that the function $f(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^2}$ is analytic in the unit disk $|z| < 1$ and find its derivative.
6. Use the Ratio Test to find the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{2^n}{n!} (z-i)^n$.
7. Expand the function $f(z) = \frac{1}{\sin(z)}$ as a Laurent series around $z_0 = 0$ and determine the residue at $z = 0$.
8. Find a conformal map that transforms the upper half-plane $\{z \in \mathbb{C} : \text{Im}(z) > 0\}$ onto the unit disk $\{z \in \mathbb{C} : |z| < 1\}$.