6.3 Taylor and Laurent series

Taylor and Laurent series are powerful tools for representing complex functions. They allow us to express analytic functions as infinite sums of powers, providing insights into their behavior and properties.

These series expansions are crucial for understanding singularities, convergence, and analytic continuation. They form the foundation for many important theorems and applications in complex analysis, bridging the gap between local and global function properties.

Taylor Series Expansions

Power Series Representation of Analytic Functions

• A function $f(z)$ is analytic at a point $z_0$ if it has a power series expansion in some neighborhood of $z_0$
  • This power series representation is called the Taylor series of $f$ centered at $z_0$
• The Taylor series of an analytic function $f(z)$ centered at $z_0$ is given by $f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n$, where $f^{(n)}(z_0)$ denotes the $n$th derivative of $f$ evaluated at $z_0$
  • The coefficients of the Taylor series, $\frac{f^{(n)}(z_0)}{n!}$, are uniquely determined by the function $f$ and the center point $z_0$
  • They can be found by repeatedly differentiating $f$ and evaluating at $z_0$

Convergence and Representation of Taylor Series

• The radius of convergence of the Taylor series is the largest radius $R$ such that the series converges for all $z$ satisfying $|z - z_0| < R$
  • Within this radius, the Taylor series represents the original function $f(z)$
• If a function is analytic at every point within a disk centered at $z_0$, then its Taylor series centered at $z_0$ converges to the function at every point in the disk
  • Example: The Taylor series of $e^z$ centered at $z_0 = 0$ converges to $e^z$ for all $z$ in the complex plane
  • Example: The Taylor series of $\sin(z)$ centered at $z_0 = 0$ converges to $\sin(z)$ for all $z$ in the complex plane

Taylor Series for Elementary Functions

Exponential, Trigonometric, and Logarithmic Functions

• The Taylor series of the exponential function $e^z$ centered at $z_0 = 0$ is given by $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$
  • This series converges for all $z$ in the complex plane
• The Taylor series of the sine function $\sin(z)$ centered at $z_0 = 0$ is given by $\sin(z) = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!}$
  • This series converges for all $z$ in the complex plane
• The Taylor series of the cosine function $\cos(z)$ centered at $z_0 = 0$ is given by $\cos(z) = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!}$
  • This series converges for all $z$ in the complex plane
• The Taylor series of the logarithmic function $\ln(1+z)$ centered at $z_0 = 0$ is given by $\ln(1+z) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} z^n}{n}$
  • This series converges for $|z| < 1$

Binomial Function

• The Taylor series of the binomial function $(1+z)^{\alpha}$ centered at $z_0 = 0$ is given by $(1+z)^{\alpha} = \sum_{n=0}^{\infty} \binom{\alpha}{n} z^n$, where $\binom{\alpha}{n} = \frac{\alpha (\alpha-1) \cdots (\alpha-n+1)}{n!}$
  • This series converges for $|z| < 1$
  • Example: The Taylor series of $\sqrt{1+z} = (1+z)^{1/2}$ centered at $z_0 = 0$ is given by $\sqrt{1+z} = 1 + \frac{1}{2}z - \frac{1}{8}z^2 + \frac{1}{16}z^3 - \cdots$

Laurent Series Expansions

Representation and Convergence of Laurent Series

• A Laurent series is a generalization of a Taylor series that allows for negative powers of $(z - z_0)$
  • It is used to represent functions in annular domains, which are regions bounded by two concentric circles
• The Laurent series of a function $f(z)$ in an annular domain centered at $z_0$ is given by $f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n$, where the coefficients $a_n$ are complex numbers
• The Laurent series of a function $f(z)$ converges in an annular domain $R_1 < |z - z_0| < R_2$, where $R_1$ and $R_2$ are the inner and outer radii of convergence, respectively

Calculating Laurent Series Coefficients

• The coefficients $a_n$ of the Laurent series can be found using the integral formula: $a_n = \frac{1}{2\pi i} \oint_C \frac{f(z)}{(z - z_0)^{n+1}} dz$, where $C$ is any closed path within the annular domain that encircles $z_0$ once counterclockwise
• If a function has a Laurent series with no negative powers of $(z - z_0)$, then the Laurent series reduces to a Taylor series, and the function is analytic in the annular domain
  • Example: The function $f(z) = \frac{1}{z(z-1)}$ has a Laurent series expansion in the annular domain $0 < |z| < 1$ given by $f(z) = \frac{1}{z} + \sum_{n=0}^{\infty} z^n$

Singular Point Classification

Types of Singular Points

• Singular points are points where a function is not analytic
  • The behavior of a function near a singular point can be characterized using its Laurent series expansion
• An isolated singular point $z_0$ is a removable singularity if the Laurent series of $f(z)$ at $z_0$ has no negative power terms (i.e., $a_n = 0$ for all $n < 0$)
  • In this case, the function can be made analytic at $z_0$ by defining $f(z_0) = a_0$
  • Example: The function $f(z) = \frac{\sin(z)}{z}$ has a removable singularity at $z_0 = 0$
• An isolated singular point $z_0$ is a pole of order $m$ if the Laurent series of $f(z)$ at $z_0$ has a finite number of negative power terms, with the lowest power being $(z - z_0)^{-m}$
  • The function approaches infinity as $z$ approaches $z_0$
  • A pole of order 1 is called a simple pole
  • Example: The function $f(z) = \frac{1}{(z-1)^2}$ has a pole of order 2 at $z_0 = 1$
• An isolated singular point $z_0$ is an essential singularity if the Laurent series of $f(z)$ at $z_0$ has an infinite number of negative power terms
  • The function exhibits complex behavior near $z_0$, such as oscillating rapidly or approaching different limits along different paths
  • Example: The function $f(z) = e^{\frac{1}{z}}$ has an essential singularity at $z_0 = 0$

Residues and the Residue Theorem

• The residue of a function $f(z)$ at an isolated singular point $z_0$ is the coefficient $a_{-1}$ of the $(z - z_0)^{-1}$ term in the Laurent series expansion of $f(z)$ at $z_0$
• Residues are used in the residue theorem for evaluating complex integrals
  • The residue theorem states that if $f(z)$ is analytic in a simply connected domain $D$ except for a finite number of isolated singular points $z_1, z_2, \ldots, z_n$, then $\oint_C f(z) dz = 2\pi i \sum_{k=1}^{n} \text{Res}(f, z_k)$, where $C$ is a closed path in $D$ that does not pass through any of the singular points and $\text{Res}(f, z_k)$ denotes the residue of $f$ at $z_k$