5.2 Power series

Power series are fundamental tools in complex analysis, representing functions as infinite sums of terms with increasing powers. They provide a way to study function behavior, convergence, and analytic properties within specific regions of the complex plane.

Understanding power series is crucial for grasping key concepts in complex analysis, including Taylor and Laurent expansions, analytic continuation, and solving differential equations. These series form the foundation for exploring the local and global properties of complex functions.

Definition of power series

• Power series are infinite series where each term is a constant multiplied by a variable raised to a non-negative integer power
• The general form of a power series is $\sum_{n=0}^{\infty} a_n (z-c)^n$, where $a_n$ are complex coefficients, $z$ is a complex variable, and $c$ is the center of the series

Formal definition

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• A power series centered at $c$ is defined as $\sum_{n=0}^{\infty} a_n (z-c)^n$, where $a_n \in \mathbb{C}$ and $z, c \in \mathbb{C}$
• The series converges for values of $z$ within a certain radius around the center $c$, called the radius of convergence
• Outside the radius of convergence, the series may diverge or converge conditionally

Center and radius of convergence

• The center of a power series is the point $c$ around which the series is expanded
• The radius of convergence $R$ is the distance from the center within which the series converges absolutely
• The series converges absolutely for $|z-c| < R$, may converge conditionally for $|z-c| = R$, and diverges for $|z-c| > R$

Convergence of power series

• The convergence of a power series depends on the values of $z$ and the coefficients $a_n$
• Power series can exhibit absolute convergence, conditional convergence, or divergence

Absolute vs conditional convergence

• A power series $\sum_{n=0}^{\infty} a_n (z-c)^n$ converges absolutely if $\sum_{n=0}^{\infty} |a_n (z-c)^n|$ converges
• Absolute convergence implies convergence, but the converse is not always true
• A series may converge conditionally if it converges but not absolutely (alternating series)

Cauchy-Hadamard theorem

• The Cauchy-Hadamard theorem provides a formula for calculating the radius of convergence $R$ of a power series
• $R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}}$, where $a_n$ are the coefficients of the power series
• If $\limsup_{n \to \infty} \sqrt[n]{|a_n|} = 0$, the series converges for all $z \in \mathbb{C}$ (entire function)

Abel's theorem on power series

• Abel's theorem states that if a power series converges at a point on its circle of convergence, then it converges uniformly on any compact subset of the open disk of convergence
• This theorem ensures the continuity of the sum function up to the boundary of the disk of convergence
• Abel's theorem is crucial for understanding the behavior of power series on the boundary of their convergence disks

Properties of power series

• Power series possess several important properties that make them useful in complex analysis
• These properties include uniqueness, term-by-term differentiation and integration, and composition

Uniqueness of power series

• If two power series $\sum_{n=0}^{\infty} a_n (z-c)^n$ and $\sum_{n=0}^{\infty} b_n (z-c)^n$ converge to the same function in a common disk of convergence, then $a_n = b_n$ for all $n$
• This property ensures that a function can be represented by only one power series in a given disk of convergence
• Uniqueness is essential for the study of analytic functions and their local behavior

Term-by-term differentiation and integration

• If a power series $\sum_{n=0}^{\infty} a_n (z-c)^n$ converges in a disk $|z-c| < R$, then it can be differentiated or integrated term-by-term within the same disk
• The differentiated series is $\sum_{n=1}^{\infty} na_n (z-c)^{n-1}$, and the integrated series is $\sum_{n=0}^{\infty} \frac{a_n}{n+1} (z-c)^{n+1} + C$
• Term-by-term differentiation and integration preserve the radius of convergence but may change the behavior on the boundary

Composition of power series

• If $f(z) = \sum_{n=0}^{\infty} a_n (z-c)^n$ and $g(z) = \sum_{n=0}^{\infty} b_n (z-d)^n$ are two power series with radii of convergence $R_f$ and $R_g$ respectively, then the composition $f(g(z))$ is also a power series
• The composed series has a radius of convergence $R \geq \min\{R_f, R_g\}$ and can be computed using the Cauchy product formula
• Composition of power series is useful for studying the behavior of analytic functions under composition and for finding power series representations of composite functions

Taylor series

• Taylor series are power series expansions of functions around a specific point
• They provide local approximations of functions and are essential for studying the behavior of analytic functions

Definition and properties

• The Taylor series of a function $f(z)$ around a point $c$ is given by $\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (z-c)^n$, where $f^{(n)}(c)$ denotes the $n$-th derivative of $f$ at $c$
• If a function $f(z)$ is analytic in a disk centered at $c$, then its Taylor series converges to $f(z)$ within that disk
• The Taylor series of a function is unique, and the coefficients are determined by the derivatives of the function at the center

Taylor series for elementary functions

• Many elementary functions have well-known Taylor series expansions
• The exponential function $e^z$ has the Taylor series $\sum_{n=0}^{\infty} \frac{z^n}{n!}$, which converges for all $z \in \mathbb{C}$
• The trigonometric functions $\sin(z)$ and $\cos(z)$ have Taylor series that converge for all $z \in \mathbb{C}$, with alternating terms and factorial denominators

Maclaurin series

• A Maclaurin series is a special case of a Taylor series centered at $c=0$
• The Maclaurin series of a function $f(z)$ is given by $\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} z^n$
• Maclaurin series are often used to study the behavior of functions near the origin and to find power series representations of functions

Laurent series

• Laurent series are a generalization of power series that allow for negative integer powers of the variable
• They are used to study the behavior of functions in annular regions and to classify singularities

Definition and properties

• A Laurent series centered at $c$ is defined as $\sum_{n=-\infty}^{\infty} a_n (z-c)^n$, where $a_n \in \mathbb{C}$ and $z, c \in \mathbb{C}$
• The series converges in an annular region $R_1 < |z-c| < R_2$, where $R_1$ and $R_2$ are the inner and outer radii of convergence
• Laurent series can be uniquely decomposed into a principal part (negative powers) and an analytic part (non-negative powers)

Laurent expansion

• The Laurent expansion of a function $f(z)$ around a point $c$ is the Laurent series representation of $f(z)$ in an annular region centered at $c$
• To find the Laurent expansion, one can use the Cauchy integral formula or manipulate the Taylor series of the function
• Laurent expansions are useful for studying the behavior of functions near isolated singularities and for computing residues

Principal part and analytic part

• The principal part of a Laurent series $\sum_{n=-\infty}^{\infty} a_n (z-c)^n$ is the sum of terms with negative powers of $(z-c)$, i.e., $\sum_{n=-\infty}^{-1} a_n (z-c)^n$
• The analytic part is the sum of terms with non-negative powers of $(z-c)$, i.e., $\sum_{n=0}^{\infty} a_n (z-c)^n$, which is a power series
• The principal part determines the type of singularity at $c$ (pole, essential singularity, or removable singularity), while the analytic part determines the behavior of the function near $c$

Applications of power series

• Power series have numerous applications in complex analysis and other fields of mathematics
• They are used for analytic continuation, evaluation of definite integrals, and solving differential equations

Analytic continuation

• Analytic continuation is the process of extending the domain of an analytic function beyond its initial region of definition
• Power series can be used to perform analytic continuation by considering the series as a new function defined on a larger domain
• Analytic continuation is important for studying the global behavior of complex functions and for discovering connections between seemingly different functions

Evaluation of definite integrals

• Power series can be used to evaluate definite integrals that are difficult or impossible to compute using standard techniques
• By expanding the integrand as a power series and integrating term-by-term, one can obtain a series representation of the integral
• This method is particularly useful for integrals involving trigonometric, exponential, or logarithmic functions

Solving differential equations

• Power series can be used to solve linear differential equations with variable coefficients
• By assuming a power series solution and substituting it into the differential equation, one can determine the coefficients of the series
• This method is called the power series method or the Frobenius method and is effective for solving differential equations around ordinary or regular singular points

Relationship with other concepts

• Power series are closely related to other important concepts in complex analysis
• Understanding these relationships is crucial for a comprehensive grasp of the subject

Power series vs Taylor series

• Power series are more general than Taylor series, as they allow for arbitrary centers of expansion
• Taylor series are a special case of power series, where the coefficients are determined by the derivatives of the function at the center
• Every Taylor series is a power series, but not every power series is a Taylor series

Power series and complex differentiability

• If a function $f(z)$ is represented by a power series in a disk centered at $c$, then $f(z)$ is analytic (complex differentiable) in that disk
• The converse is also true: if $f(z)$ is analytic in a disk, then it can be represented by a power series in that disk
• Power series provide a convenient way to study the complex differentiability of functions and to prove that certain functions are analytic

Power series and residue theory

• Residue theory is a powerful tool in complex analysis that allows for the evaluation of certain integrals using the residues of a function at its poles
• Laurent series are essential for computing residues, as the residue of a function at a pole is the coefficient of the $\frac{1}{z-c}$ term in its Laurent expansion
• Power series and Laurent series are fundamental for understanding and applying residue theory to solve problems in complex analysis and other fields