Intro to Complex Analysis

💠Intro to Complex Analysis Unit 5 – Analytic Function Series Representations

Analytic function series representations are a cornerstone of complex analysis. They allow us to express complex functions as infinite sums, providing powerful tools for studying their behavior and properties. These representations include power series, Taylor series, and Laurent series. Understanding these series is crucial for analyzing singularities, evaluating complex integrals, and exploring the convergence properties of functions. They form the foundation for many advanced techniques in complex analysis, such as residue theory and conformal mapping.

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 n=0an(zz0)n\sum_{n=0}^{\infty} a_n (z-z_0)^n, where ana_n are complex coefficients and z0z_0 is the center of the series
    • The series converges for values of zz within a certain radius of convergence
  • Taylor series are power series expansions of analytic functions around a point z0z_0, representing the function as an infinite sum of terms involving its derivatives at z0z_0
  • Laurent series are power series expansions that allow for negative powers of (zz0)(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 (zz0)(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 n=0an(zz0)n\sum_{n=0}^{\infty} a_n (z-z_0)^n, where ana_n are complex coefficients and z0z_0 is the center of the series
  • The series converges for values of zz within a certain radius of convergence, denoted by RR, 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=limnanan+1R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|
    • Root test: R=1limnannR = \frac{1}{\lim_{n \to \infty} \sqrt[n]{|a_n|}}
  • Examples of power series include the geometric series n=0zn\sum_{n=0}^{\infty} z^n (converges for z<1|z| < 1) and the exponential series n=0znn!\sum_{n=0}^{\infty} \frac{z^n}{n!} (converges for all zz)

Taylor Series for Analytic Functions

  • Taylor series are power series expansions of analytic functions around a point z0z_0, representing the function as an infinite sum of terms involving its derivatives at z0z_0
  • The Taylor series of an analytic function f(z)f(z) around z0z_0 is given by f(z)=n=0f(n)(z0)n!(zz0)nf(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z-z_0)^n
    • f(n)(z0)f^{(n)}(z_0) denotes the nn-th derivative of ff evaluated at z0z_0
  • If the Taylor series converges to f(z)f(z) for all zz within some disk centered at z0z_0, the function is said to be analytic at z0z_0
  • The error in approximating f(z)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 z0=0z_0 = 0) for eze^z, sin(z)\sin(z), and cos(z)\cos(z)
    • ez=n=0znn!e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}
    • sin(z)=n=0(1)nz2n+1(2n+1)!\sin(z) = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!}
    • cos(z)=n=0(1)nz2n(2n)!\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 (zz0)(z-z_0), useful for studying the behavior of functions near singularities
  • The Laurent series of a function f(z)f(z) around z0z_0 is given by f(z)=n=an(zz0)nf(z) = \sum_{n=-\infty}^{\infty} a_n (z-z_0)^n
    • The series converges in an annular region centered at z0z_0, with inner radius R1R_1 and outer radius R2R_2
  • Singularities are classified based on the Laurent series expansion:
    • Removable singularity: an=0a_n = 0 for all n<0n < 0, and the function can be redefined to be analytic at z0z_0
    • Pole of order mm: an=0a_n = 0 for all n<mn < -m, and the function behaves like (zz0)m(z-z_0)^{-m} near z0z_0
    • Essential singularity: an0a_n \neq 0 for infinitely many negative nn, and the function exhibits complex behavior near z0z_0
  • The residue of a function at a pole is the coefficient a1a_{-1} in its Laurent series expansion
    • Residues are useful for evaluating complex integrals using the Residue Theorem
  • Examples of functions with singularities include 1z\frac{1}{z} (pole at z=0z=0), 1sin(z)\frac{1}{\sin(z)} (poles at z=nπz=n\pi), and e1ze^{\frac{1}{z}} (essential singularity at z=0z=0)

Convergence and Radius of Convergence

  • The convergence of a power series n=0an(zz0)n\sum_{n=0}^{\infty} a_n (z-z_0)^n depends on the values of zz and the behavior of the coefficients ana_n
  • The radius of convergence RR is the largest radius of a disk centered at z0z_0 within which the series converges
    • The series converges absolutely for zz0<R|z-z_0| < R, converges conditionally for zz0=R|z-z_0| = R (in some cases), and diverges for zz0>R|z-z_0| > R
  • The ratio test and the root test are commonly used to determine the radius of convergence
    • Ratio test: R=limnanan+1R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|
    • Root test: R=1limnannR = \frac{1}{\lim_{n \to \infty} \sqrt[n]{|a_n|}}
  • For Taylor series, the radius of convergence is the distance from the center z0z_0 to the nearest singularity of the function
  • The convergence of Laurent series is determined by the inner and outer radii, R1R_1 and R2R_2, which can be found using the ratio or root test on the positive and negative power terms separately
  • Examples:
    • The geometric series n=0zn\sum_{n=0}^{\infty} z^n has a radius of convergence R=1R=1
    • The series n=1znn2\sum_{n=1}^{\infty} \frac{z^n}{n^2} has a radius of convergence R=1R=1 (ratio test)

Applications in Complex Integration

  • Cauchy's Integral Formula expresses the value of an analytic function f(z)f(z) at a point z0z_0 in terms of a contour integral: f(z0)=12πiCf(z)zz0dzf(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-z_0} dz
    • The contour CC must enclose z0z_0 and lie within a region where f(z)f(z) is analytic
  • The Residue Theorem states that for a meromorphic function f(z)f(z) and a simple closed contour CC, Cf(z)dz=2πik=1nRes(f,zk)\oint_C f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)
    • Res(f,zk)\text{Res}(f, z_k) is the residue of f(z)f(z) at the pole zkz_k, and the sum is taken over all poles enclosed by CC
  • 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)f(z) is analytic in a region and continuous on its boundary, then f(z)|f(z)| attains its maximum value on the boundary
  • Examples:
    • Evaluating sin(x)xdx\int_{-\infty}^{\infty} \frac{\sin(x)}{x} dx using a semicircular contour in the upper half-plane
    • Computing 02πdθ2+cos(θ)\int_0^{2\pi} \frac{d\theta}{2+\cos(\theta)} using the Residue Theorem with the substitution z=eiθz=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., z21z1\frac{z^2-1}{z-1} at z=1z=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)=11zf(z) = \frac{1}{1-z} around z0=0z_0 = 0 and determine its radius of convergence.
  2. Classify the singularities of f(z)=ez(z1)2(z+i)f(z) = \frac{e^z}{(z-1)^2(z+i)} and find the residue at each pole.
  3. Evaluate the integral 0xsin(x)x2+1dx\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)=(z2)2(z+1)(zi)3f(z) = \frac{(z-2)^2(z+1)}{(z-i)^3} inside the unit circle z=1|z| = 1 using the Argument Principle.
  5. Find the Laurent series expansion of f(z)=1z(z1)f(z) = \frac{1}{z(z-1)} in the annular region 0<z<10 < |z| < 1 and determine the residue at z=0z=0.
  6. Prove that the function f(z)=ezf(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 n=1(zi)nn3\sum_{n=1}^{\infty} \frac{(z-i)^n}{n^3}.
  8. Apply the Maximum Modulus Principle to show that the function f(z)=z2f(z) = z^2 attains its maximum value on the boundary of the unit disk z1|z| \leq 1.


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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