Power Series Representations

Welcome, students. In this lesson, you will learn how power series turn complicated complex functions into sums that behave like polynomials, why that is useful, and how this idea connects to Taylor series, Midterm 1, and analytic continuation ✨. By the end, you should be able to explain what a power series representation is, recognize when one exists, and use it to compute or understand a function near a point.

Why Power Series Matter

A power series is a function written as an infinite polynomial-shaped sum:

$\sum_{n=0}^{\infty} a_n (z-z_0)^n$

Here, $z$ is a complex variable, $z_0$ is the center, and the numbers $a_n$ are coefficients. This looks like a polynomial, but it may have infinitely many terms. That infinite structure is what makes power series extremely powerful in complex analysis 📘.

Why do we care? Because many complex functions are much easier to study when they are written as power series. Around a point where the series converges, the function behaves like a polynomial. That means you can approximate values, study derivatives, and understand local behavior using algebraic methods.

For example, near $z=0$, the geometric series

$\frac{1}{1-z} = \sum_{n=0}^{\infty} z^n$

is a power series representation valid when $|z|<1$. Instead of working directly with a fraction, we can work with the sum of simple terms. This is often easier for analysis, computation, and proof.

Main terms to know

A power series is a series of the form $\sum_{n=0}^{\infty} a_n (z-z_0)^n$.
A center is the point $z_0$ around which the series is built.
A radius of convergence is the number $R$ such that the series converges when $|z-z_0|<R$ and diverges when $|z-z_0|>R$.
The disk of convergence is the set $\{z: |z-z_0|<R\}$.

These ideas are central in Midterm 1 because they connect algebra, limits, and function behavior in the complex plane.

How Convergence Works

Not every power series converges everywhere. In fact, every power series has a convergence pattern determined by its coefficients. For a series

$\sum_{n=0}^{\infty} a_n (z-z_0)^n,$

there are three possibilities for each point $z$:

it converges absolutely,
it diverges,
or it sits on the boundary, where special tests may be needed.

A key theorem says there is always a radius $R$ such that the series converges for $|z-z_0|<R$ and diverges for $|z-z_0|>R$. The boundary case $|z-z_0|=R$ must be checked separately.

A useful way to estimate $R$ is the ratio test. If the limit exists, then for the coefficients of a power series, one may examine

$\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|$

and related expressions to determine where the terms shrink fast enough. Another common tool is the root test using

$\limsup_{n\to\infty} |a_n|^{1/n}.$

For students, the important idea is this: the coefficients control where the series is valid. If the coefficients grow too quickly, the radius may be small. If they shrink fast enough, the radius may be large or even infinite.

Example: geometric series

The geometric series is the model example:

$\sum_{n=0}$^{$\infty$} z^n = 1 + z + z^2 + z^3 + $\cdots$

This converges when $|z|<1$, and its sum is

$\frac{1}{1-z}.$

This identity is one of the most important power series representations in complex analysis because it can be transformed into many others by substitution and algebra.

For instance, replacing $z$ by $-z$ gives

$\frac{1}{1+z}$ = $\sum_{n=0}$^{$\infty$} (-1)^n z^n, \quad |z|<1.

Replacing $z$ by $z^2$ gives

$\frac{1}{1-z^2} = \sum_{n=0}^{\infty} z^{2n}, \quad |z|<1.$

These are simple but powerful examples of how a single known series can generate many others.

Taylor Series as Power Series Representations

A Taylor series is a special power series representation of a function around a point $z_0$. If a function $f$ has derivatives of all orders near $z_0$, its Taylor series is

$\sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n.$

This formula is the complex-analytic version of the same idea from calculus, but in complex analysis the result is much stronger. For many functions that are analytic, the Taylor series actually equals the function inside some disk.

A function is analytic at a point if it can be represented by a power series in a neighborhood of that point. This means the function is not just infinitely differentiable there; it is locally identical to its power series. That is a major idea in Complex Analysis.

Example: exponential function

The complex exponential has the Taylor series

$ e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}.$

This series converges for every complex number $z$, so its radius of convergence is infinite. That means $e^z$ has a power series representation valid everywhere in the complex plane.

This is a great example because the same series defines the function for both real and complex inputs. It also shows how power series can extend familiar functions beyond what you may first see in algebra.

Example: sine and cosine

The trigonometric functions also have global power series representations:

$\sin z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!},$

$\cos z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!}.$

These formulas are useful because they define $\sin z$ and $\cos z$ for complex inputs and connect directly to differentiation rules.

How to Build a Power Series Representation

There are several standard ways to find a power series representation.

1. Start with a known series

Use the geometric series and manipulate it. For example,

$\frac{1}{2-z} = \frac{1}{2}\cdot \frac{1}{1-\frac{z}{2}}.$

Now apply the geometric series formula:

$\frac{1}{2-z}$ = $\frac{1}{2}$$\sum_{n=0}$^{$\infty$} $\left($$\frac{z}{2}$$\right)$^n = $\sum_{n=0}$^{$\infty$} $\frac{z^n}{2^{n+1}}$, \quad |z|<2.

This method is especially common on exams because it is fast and reliable when the function is algebraic.

2. Differentiate or integrate a known series

If a power series converges inside a disk, then you can differentiate or integrate it term by term inside that disk. For example, from

$\sum_{n=0}^{\infty} z^n = \frac{1}{1-z},$

differentiate both sides to get

$\sum_{n=1}$^{$\infty$} n z^{n-1} = $\frac{1}{(1-z)^2}$, \quad |z|<1.

You can also write

$\sum_{n=0}^{\infty} (n+1) z^n = \frac{1}{(1-z)^2}.$

This is an important procedure in Complex Analysis and a frequent tool in Midterm 1 problems.

3. Match a Taylor formula

If the function is analytic and derivatives are manageable, compute $f^{(n)}(z_0)$ and plug into

$\sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n.$

This method is especially useful for functions like $e^z$, $\sin z$, and $\cos z$.

Analytic Continuation Intuition

Power series representations also help explain analytic continuation. The basic idea is that if a function is given by a power series on one disk, it may sometimes be extended to a larger region by finding another power series that agrees with it on an overlap.

Imagine you know a function near one point, like $z_0=0$, and you want to understand it farther away. If you can find another power series centered at a different point that matches the same function on a shared region, then you have extended the function’s description. This is the intuition behind analytic continuation 🔎.

A simple example is $\frac{1}{1-z}$. Its power series about $z=0$ converges only for $|z|<1$, but the function itself is defined for all $z\neq 1$. The power series does not cover the whole function, yet the function exists beyond the disk. This shows the difference between a power series representation and the underlying function.

In complex analysis, analytic continuation is often possible because analytic functions are highly rigid: once you know them on a small region, their behavior is strongly constrained elsewhere. That is one reason power series are so important.

Conclusion

Power series representations are one of the core tools in Complex Analysis because they turn analytic functions into infinite polynomial-like expressions. students, you should now understand the key terms: center, coefficients, radius of convergence, and analytic function. You should also know how power series connect to Taylor series, how to build them from known series, and why they matter for extending functions across regions of the complex plane.

On Midterm 1, be ready to recognize a power series, determine where it converges, convert a function into a series, and explain why the series represents the function inside its disk of convergence. These skills are foundational for the rest of Taylor series and for later ideas like analytic continuation.

Study Notes

A power series has the form $\sum_{n=0}^{\infty} a_n (z-z_0)^n$.
The point $z_0$ is the center of the series.
The radius of convergence $R$ gives the disk $|z-z_0|<R$ where the series converges.
Boundary points with $|z-z_0|=R$ must be checked separately.
A Taylor series is a power series with coefficients $\frac{f^{(n)}(z_0)}{n!}$.
If a function is analytic, it can be represented by a power series near the point.
The geometric series $\sum_{n=0}^{\infty} z^n = \frac{1}{1-z}$ is the main template for building many other series.
Differentiating or integrating a convergent power series term by term is a standard technique.
Common examples include $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$, $\sin z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!}$, and $\cos z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!}$.
Power series representations help explain analytic continuation by extending a function from one region to another through overlapping series.