Power Series Representations

Why This Matters

Power series are the backbone of complex analysis—they transform complicated functions into infinite sums that you can manipulate, differentiate, and integrate term by term. You're being tested on your ability to recognize when a function can be represented as a power series, where that representation is valid (convergence), and how to use these expansions to analyze function behavior near specific points. These concepts connect directly to analyticity, residue calculations, and contour integration.

Don't just memorize the formulas for each series. Know why Taylor series center matters, how radius of convergence limits your analysis, and what distinguishes entire functions from those with singularities. When an exam asks you to expand a function or evaluate a limit, you need to recognize which series representation applies and where it's valid—that's the real skill being tested.

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Foundational Series Structures

The Taylor and Maclaurin series provide the fundamental framework for representing analytic functions as power series. Any function that's analytic at a point can be expressed as a convergent power series in some neighborhood of that point.

Taylor Series

• Represents any analytic function as an infinite sum centered at point $a$, with the general form $f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (z - a)^n$
• Coefficients encode all derivative information—the $n$th coefficient is $\frac{f^{(n)}(a)}{n!}$, connecting local behavior to global representation
• Convergence occurs within a disk centered at $a$, extending to the nearest singularity in the complex plane

Maclaurin Series

• Special case of Taylor series with $a = 0$—simplifies to $f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} z^n$
• Default choice for functions analytic at the origin—most standard series you'll memorize are Maclaurin series
• Computational efficiency comes from evaluating derivatives at zero, which often produces clean patterns

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Building Block Series

These fundamental series serve as templates for deriving more complex expansions. Master these forms and you can construct most series you'll encounter through substitution and manipulation.

Geometric Series

• Converges to $\frac{a}{1-r}$ when $|r| < 1$—the series $\sum_{n=0}^{\infty} ar^n$ is the prototype for understanding convergence
• Foundation for rational function expansions—rewrite expressions as $\frac{1}{1-(\text{something})}$ to generate series
• Radius of convergence is exactly 1 for the standard form, determined by the singularity at $r = 1$

Binomial Series

• Generalizes $(1+z)^k$ to non-integer exponents—given by $\sum_{n=0}^{\infty} \binom{k}{n} z^n$ where $\binom{k}{n} = \frac{k(k-1)\cdots(k-n+1)}{n!}$
• Converges for $|z| < 1$ when $k$ is not a non-negative integer; terminates for non-negative integer $k$
• Essential for fractional powers and roots—expanding $\sqrt{1+z}$ or $(1+z)^{-1}$ requires this series

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Entire Function Series

These series converge everywhere in the complex plane, representing entire functions—functions analytic on all of $\mathbb{C}$ with no singularities.

Exponential Function Series

• Converges for all $z \in \mathbb{C}$—the series $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$ has infinite radius of convergence
• Factorial growth in denominators ensures convergence regardless of how large $|z|$ becomes
• Connects to Euler's formula $e^{iz} = \cos z + i\sin z$, linking exponential and trigonometric series

Sine and Cosine Series

• Sine uses only odd powers—$\sin z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!}$ reflects the odd symmetry of sine
• Cosine uses only even powers—$\cos z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!}$ reflects the even symmetry of cosine
• Both converge for all $z$ and can be derived from the exponential series using $e^{iz}$

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Series with Finite Convergence

These series have restricted domains of convergence, typically due to singularities that limit how far the power series can extend.

Logarithmic Series

• Converges only for $|z| < 1$—the series $\ln(1+z) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} z^n}{n}$ is limited by the branch point at $z = -1$
• Starts at $n = 1$, not $n = 0$—there's no constant term since $\ln(1) = 0$
• Slower convergence than factorial-based series due to the $\frac{1}{n}$ coefficients rather than $\frac{1}{n!}$

Radius of Convergence

• Measures the disk where the series converges absolutely—for $\sum a_n (z-a)^n$, convergence holds when $|z-a| < R$
• Calculated via ratio test $R = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right|$ or root test $R = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}}$
• Determined by nearest singularity in the complex plane—the series cannot converge past any point where $f(z)$ fails to be analytic

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Advanced Convergence Theory

These results govern the behavior of power series at boundary points and extend representations to functions with singularities.

Abel's Theorem

• Extends convergence to boundary points—if $\sum a_n$ converges, then $\lim_{x \to 1^-} \sum a_n x^n = \sum a_n$
• Guarantees continuity at convergent boundary points—the function defined by the series is continuous from within the disk
• Critical for summing conditionally convergent series like $\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \cdots$

Laurent Series

• Includes negative powers of $(z-a)$—the general form $f(z) = \sum_{n=-\infty}^{\infty} a_n (z-a)^n$ handles singularities
• Converges in an annulus $r < |z-a| < R$, not a disk—the inner radius excludes the singularity
• Essential for residue calculation—the coefficient $a_{-1}$ is the residue at $z = a$, fundamental to contour integration

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Quick Reference Table

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Self-Check Questions

1. Which two series are entire functions that can be combined to derive the third via Euler's formula? What is that relationship?

2. A function has a simple pole at $z = 2$. If you expand it in a Taylor series centered at $z = 0$, what is the maximum possible radius of convergence, and why?

3. Compare and contrast the logarithmic series and the geometric series: what limits the convergence of each, and how do their coefficient patterns differ?

4. You need to expand $f(z) = \frac{1}{(z-1)(z-3)}$ around $z = 0$. Which fundamental series form would you use, and what's the radius of convergence?

5. When would you choose a Laurent series over a Taylor series? Give a specific scenario involving residue calculation where this choice is necessary.