💠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.
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=0∞an(z−z0)n, where an are complex coefficients and z0 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 z0, representing the function as an infinite sum of terms involving its derivatives at z0
Laurent series are power series expansions that allow for negative powers of (z−z0), 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−z0)
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=0∞an(z−z0)n, where an are complex coefficients and z0 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=limn→∞an+1an
Root test: R=limn→∞n∣an∣1
Examples of power series include the geometric series ∑n=0∞zn (converges for ∣z∣<1) and the exponential series ∑n=0∞n!zn (converges for all z)
Taylor Series for Analytic Functions
Taylor series are power series expansions of analytic functions around a point z0, representing the function as an infinite sum of terms involving its derivatives at z0
The Taylor series of an analytic function f(z) around z0 is given by f(z)=∑n=0∞n!f(n)(z0)(z−z0)n
f(n)(z0) denotes the n-th derivative of f evaluated at z0
If the Taylor series converges to f(z) for all z within some disk centered at z0, the function is said to be analytic at z0
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 z0=0) for ez, sin(z), and cos(z)
ez=∑n=0∞n!zn
sin(z)=∑n=0∞(−1)n(2n+1)!z2n+1
cos(z)=∑n=0∞(−1)n(2n)!z2n
Laurent Series and Singularities
Laurent series are power series expansions that allow for negative powers of (z−z0), useful for studying the behavior of functions near singularities
The Laurent series of a function f(z) around z0 is given by f(z)=∑n=−∞∞an(z−z0)n
The series converges in an annular region centered at z0, with inner radius R1 and outer radius R2
Singularities are classified based on the Laurent series expansion:
Removable singularity: an=0 for all n<0, and the function can be redefined to be analytic at z0
Pole of order m: an=0 for all n<−m, and the function behaves like (z−z0)−m near z0
Essential singularity: an=0 for infinitely many negative n, and the function exhibits complex behavior near z0
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 z1 (pole at z=0), sin(z)1 (poles at z=nπ), and ez1 (essential singularity at z=0)
Convergence and Radius of Convergence
The convergence of a power series ∑n=0∞an(z−z0)n depends on the values of z and the behavior of the coefficients an
The radius of convergence R is the largest radius of a disk centered at z0 within which the series converges
The series converges absolutely for ∣z−z0∣<R, converges conditionally for ∣z−z0∣=R (in some cases), and diverges for ∣z−z0∣>R
The ratio test and the root test are commonly used to determine the radius of convergence
Ratio test: R=limn→∞an+1an
Root test: R=limn→∞n∣an∣1
For Taylor series, the radius of convergence is the distance from the center z0 to the nearest singularity of the function
The convergence of Laurent series is determined by the inner and outer radii, R1 and R2, which can be found using the ratio or root test on the positive and negative power terms separately
Examples:
The geometric series ∑n=0∞zn has a radius of convergence R=1
The series ∑n=1∞n2zn 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 z0 in terms of a contour integral: f(z0)=2πi1∮Cz−z0f(z)dz
The contour C must enclose z0 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, ∮Cf(z)dz=2πi∑k=1nRes(f,zk)
Res(f,zk) is the residue of f(z) at the pole zk, 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 ∫−∞∞xsin(x)dx using a semicircular contour in the upper half-plane
Computing ∫02π2+cos(θ)dθ using the Residue Theorem with the substitution z=eiθ
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., z−1z2−1 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
Find the Taylor series expansion of f(z)=1−z1 around z0=0 and determine its radius of convergence.
Classify the singularities of f(z)=(z−1)2(z+i)ez and find the residue at each pole.
Evaluate the integral ∫0∞x2+1xsin(x)dx using the Residue Theorem.
Determine the number of zeros and poles of f(z)=(z−i)3(z−2)2(z+1) inside the unit circle ∣z∣=1 using the Argument Principle.
Find the Laurent series expansion of f(z)=z(z−1)1 in the annular region 0<∣z∣<1 and determine the residue at z=0.
Prove that the function f(z)=ez satisfies the Cauchy-Riemann equations and is thus analytic everywhere in the complex plane.
Use the ratio test to find the radius of convergence of the power series ∑n=1∞n3(z−i)n.
Apply the Maximum Modulus Principle to show that the function f(z)=z2 attains its maximum value on the boundary of the unit disk ∣z∣≤1.