💠Intro to Complex Analysis Unit 5 – 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
  • The series converges for values of $z$ within a certain radius of convergence
• Taylor series are power series expansions of analytic functions around a point $z_0$, representing the function as an infinite sum of terms involving its derivatives at $z_0$
• Laurent series are power series expansions that allow for negative powers of $(z-z_0)$, useful for studying the behavior of functions near singularities
• Singularities are points where a function is not analytic, classified as removable, poles, or essential singularities based on the behavior of the function near the point
• Cauchy's Integral Formula expresses the value of an analytic function at a point in terms of a contour integral of the function divided by $(z-z_0)$
• Residue Theorem relates the contour integral of a meromorphic function to the sum of its residues at the poles enclosed by the contour

Power Series in Complex Analysis

• Power series in complex analysis are 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
• The series converges for values of $z$ within a certain radius of convergence, denoted by $R$, and diverges outside this radius
• Inside the radius of convergence, the power series defines an analytic function
  • The function can be differentiated or integrated term by term within the radius of convergence
• The radius of convergence can be determined using the ratio test or the root test
  • Ratio test: $R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|$
  • Root test: $R = \frac{1}{\lim_{n \to \infty} \sqrt[n]{|a_n|}}$
• Examples of power series include the geometric series $\sum_{n=0}^{\infty} z^n$ (converges for $|z| < 1$) and the exponential series $\sum_{n=0}^{\infty} \frac{z^n}{n!}$ (converges for all $z$)

Taylor Series for Analytic Functions

• Taylor series are power series expansions of analytic functions around a point $z_0$, representing the function as an infinite sum of terms involving its derivatives at $z_0$
• The Taylor series of an analytic function $f(z)$ around $z_0$ is given by $f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z-z_0)^n$
  • $f^{(n)}(z_0)$ denotes the $n$-th derivative of $f$ evaluated at $z_0$
• If the Taylor series converges to $f(z)$ for all $z$ within some disk centered at $z_0$, the function is said to be analytic at $z_0$
• The error in approximating $f(z)$ by a partial sum of its Taylor series is given by the remainder term, which can be estimated using Taylor's Theorem
• Examples of Taylor series include the Maclaurin series (Taylor series around $z_0 = 0$) for $e^z$, $\sin(z)$, and $\cos(z)$
  • $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$
  • $\sin(z) = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!}$
  • $\cos(z) = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!}$

Laurent Series and Singularities

• Laurent series are power series expansions that allow for negative powers of $(z-z_0)$, useful for studying the behavior of functions near singularities
• The Laurent series of a function $f(z)$ around $z_0$ is given by $f(z) = \sum_{n=-\infty}^{\infty} a_n (z-z_0)^n$
  • The series converges in an annular region centered at $z_0$, with inner radius $R_1$ and outer radius $R_2$
• Singularities are classified based on the Laurent series expansion:
  • Removable singularity: $a_n = 0$ for all $n < 0$, and the function can be redefined to be analytic at $z_0$
  • Pole of order $m$: $a_n = 0$ for all $n < -m$, and the function behaves like $(z-z_0)^{-m}$ near $z_0$
  • Essential singularity: $a_n \neq 0$ for infinitely many negative $n$, and the function exhibits complex behavior near $z_0$
• The residue of a function at a pole is the coefficient $a_{-1}$ in its Laurent series expansion
  • Residues are useful for evaluating complex integrals using the Residue Theorem
• Examples of functions with singularities include $\frac{1}{z}$ (pole at $z=0$), $\frac{1}{\sin(z)}$ (poles at $z=n\pi$), and $e^{\frac{1}{z}}$ (essential singularity at $z=0$)

Convergence and Radius of Convergence

• The convergence of a power series $\sum_{n=0}^{\infty} a_n (z-z_0)^n$ depends on the values of $z$ and the behavior of the coefficients $a_n$
• The radius of convergence $R$ is the largest radius of a disk centered at $z_0$ within which the series converges
  • The series converges absolutely for $|z-z_0| < R$, converges conditionally for $|z-z_0| = R$ (in some cases), and diverges for $|z-z_0| > R$
• The ratio test and the root test are commonly used to determine the radius of convergence
  • Ratio test: $R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|$
  • Root test: $R = \frac{1}{\lim_{n \to \infty} \sqrt[n]{|a_n|}}$
• For Taylor series, the radius of convergence is the distance from the center $z_0$ to the nearest singularity of the function
• The convergence of Laurent series is determined by the inner and outer radii, $R_1$ and $R_2$, which can be found using the ratio or root test on the positive and negative power terms separately
• Examples:
  • The geometric series $\sum_{n=0}^{\infty} z^n$ has a radius of convergence $R=1$
  • The series $\sum_{n=1}^{\infty} \frac{z^n}{n^2}$ has a radius of convergence $R=1$ (ratio test)

Applications in Complex Integration

• Cauchy's Integral Formula expresses the value of an analytic function $f(z)$ at a point $z_0$ in terms of a contour integral: $f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-z_0} dz$
  • The contour $C$ must enclose $z_0$ and lie within a region where $f(z)$ is analytic
• The Residue Theorem states that for a meromorphic function $f(z)$ and a simple closed contour $C$, $\oint_C f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)$
  • $\text{Res}(f, z_k)$ is the residue of $f(z)$ at the pole $z_k$, and the sum is taken over all poles enclosed by $C$
• Cauchy's Integral Formula and the Residue Theorem are powerful tools for evaluating complex integrals
  • They can be used to compute integrals of real-valued functions by extending them to the complex plane and choosing appropriate contours
• The Argument Principle relates the change in the argument of an analytic function along a closed contour to the number of zeros and poles enclosed by the contour
• The Maximum Modulus Principle states that if $f(z)$ is analytic in a region and continuous on its boundary, then $|f(z)|$ attains its maximum value on the boundary
• Examples:
  • Evaluating $\int_{-\infty}^{\infty} \frac{\sin(x)}{x} dx$ using a semicircular contour in the upper half-plane
  • Computing $\int_0^{2\pi} \frac{d\theta}{2+\cos(\theta)}$ using the Residue Theorem with the substitution $z=e^{i\theta}$

Common Pitfalls and Misconceptions

• Confusing the radius of convergence with the region of convergence
  • The radius of convergence only determines the size of the disk within which the series converges, but the actual region of convergence may be different (e.g., a half-disk or an annulus)
• Misapplying the ratio or root test by using the wrong limit or forgetting to take the absolute value
• Incorrectly classifying singularities based on the Laurent series expansion
  • A function may have a removable singularity even if some negative power terms are present (e.g., $\frac{z^2-1}{z-1}$ at $z=1$)
• Misinterpreting the Residue Theorem by including poles that are not enclosed by the contour or using the wrong residue formula for higher-order poles
• Forgetting to check the continuity of a function on the boundary when applying the Maximum Modulus Principle
• Attempting to apply Cauchy's Integral Formula or the Residue Theorem to functions that are not analytic or meromorphic in the region of interest
• Confusing the Taylor series expansion of a function with its Laurent series expansion near a singularity
• Mishandling branch cuts and multi-valued functions when integrating in the complex plane

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. Classify the singularities of $f(z) = \frac{e^z}{(z-1)^2(z+i)}$ and find the residue at each pole.
3. Evaluate the integral $\int_0^{\infty} \frac{x \sin(x)}{x^2+1} dx$ using the Residue Theorem.
4. Determine the number of zeros and poles of $f(z) = \frac{(z-2)^2(z+1)}{(z-i)^3}$ inside the unit circle $|z| = 1$ using the Argument Principle.
5. Find the Laurent series expansion of $f(z) = \frac{1}{z(z-1)}$ in the annular region $0 < |z| < 1$ and determine the residue at $z=0$.
6. Prove that the function $f(z) = e^z$ satisfies the Cauchy-Riemann equations and is thus analytic everywhere in the complex plane.
7. Use the ratio test to find the radius of convergence of the power series $\sum_{n=1}^{\infty} \frac{(z-i)^n}{n^3}$.
8. Apply the Maximum Modulus Principle to show that the function $f(z) = z^2$ attains its maximum value on the boundary of the unit disk $|z| \leq 1$.