Series Expansion: Definition, Common Types

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Contents:

• What is a Series Expansion?
• General Types
• List of Common Series Expansions

What is a Series Expansion?

A series expansion is where a function is represented by a sum of powers of either:

• One of its variables,
• Another function (usually an elementary function).

For example, the natural exponential function e^x can be expanded into an infinite series:


This particular expansion is called a Taylor series.

Series expansions have a myriad of uses in a vast array of scientific areas. For example, in calculus, if you know the value of a function at a certain point (and its derivatives), you can calculate values for the whole function. Or, if you have a particularly ugly derivative or integral, you can use a series expansion to simplify the math and find an approximate solution.

General Types of Series Expansion

The most common series expansions you’ll come across are:

• Binomial series: Two binomial quantities are raised to a power and expanded. For example, (a + b)² = (a + b) * (a + b).
• Power series: Like a polynomial of infinite degree, it can be written in a few different forms. A basic example if 1 + x + x² + … + xⁿ.
• Taylor & Maclaurin Series: approximates functions with a series of polynomial functions.
• Laurent series: a way to represent a complex function as a complex power series with negative powers.

These aren’t the only tools for series expansion though. Many others exist, but they tend to be used in very specific circumstances. For example, Zernike polynomials are used in optics to calculate the shape of aberrated wavefronts in optical systems (Indiana, 2020) and Stirling series are used for approximating factorials. Others include:

• Arctangent series expansion:
• Dirichlet series: Any series of the form a_n n^-s =: f(s) ∈ ℂ. The Reimann zeta function is a famous example (McCarthy, 2018).
  
• Legendre functions of the first kind (also called Legendre polynomials), are solutions to the Legendre differential equation.
• Puiseux series: a generalization of power series that allows for negative and fractional exponents of the indeterminate T.

Common Series Expansions

References

Indiana University Bloomington. (2020). Standards for Reporting the Optical Aberrations of Eyes.
McCarthy, J. (2018). Dirichlet Series, Retrieved December 2, 2020 from: https://www.math.wustl.edu/~mccarthy/amaster-ds.pdf

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