Series expansion

Types of series expansions

There are several kinds of series expansions, listed below.

Taylor series

A Taylor series is a power series based on a function's derivatives at a single point. More specifically, if a function f: U\to\R is infinitely differentiable around a point x_0, then the Taylor series of f around this point is given by

: \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n

under the convention 0^0 := 1. The Maclaurin series of a function f is a special case of its Taylor series about x_0 = 0.

Laurent series

A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form \sum_{k = -\infty}^{\infty} c_k (z - a)^k and converges in an annulus. In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.

Dirichlet series

A general Dirichlet series is a series of the form \sum_{n = 1}^{\infty} a_ne^{-\lambda_n s}. One important special case of this is the ordinary Dirichlet series \sum_{n = 1}^{\infty}\frac{a_n}{n^s}. Used in number theory.

Fourier series

A Fourier series is an expansion of periodic functions as a sum of many sine and cosine functions. More specifically, the Fourier series of a function f(x) of period 2L is given by the expressiona_0 + \sum_{n = 1}^{\infty} \left[a_n\cos\left(\frac{n\pi x}{L}\right) + b_n\sin\left(\frac{n\pi x}{L}\right)\right]where the coefficients are given by the formulae\begin{align} a_n &:= \frac{1}{L}\int_{-L}^L f(x)\cos\left(\frac{n\pi x}{L}\right)dx, \[5pt] b_n &:= \frac{1}{L}\int_{-L}^L f(x)\sin\left(\frac{n\pi x}{L}\right)dx. \end{align}

Other series

• In acoustics, e.g., the fundamental tone and the overtones together form an example of a Fourier series.

• Newtonian series

• Legendre polynomials: Used in physics to describe an arbitrary electrical field as a superposition of a dipole field, a quadrupole field, an octupole field, etc.

• Zernike polynomials: Used in optics to calculate aberrations of optical systems. Each term in the series describes a particular type of aberration.

• The Stirling series\operatorname{Ln}\Gamma(z)\sim\left(z-\tfrac{1}{2}\right)\ln z-z+\tfrac{1}{2}\ln(2\pi)+\sum_{k=1}^\infty \frac{B_{2k}}{2k(2k-1)z^{2k-1}}is an approximation of the log-gamma function.

Examples

The following is the Taylor series of the exponential function e^x:e^x=\sum^\infty_{n=0}\frac{x^n}{n!}= 1 + x + \frac{x^2}{2} + \frac{x^3}{6}+\cdots

The Dirichlet series of the Riemann zeta function is\zeta(s) := \sum_{n = 1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \cdots

References

References

1. (2013-11-07). "Series and Expansions".
2. (2007-01-01). "Numerical Methods for Special Functions". SIAM.
3. (27 December 2013). "Taylor series - Encyclopedia of Mathematics".
4. (2008). "Elementary Differential Equations with Boundary Value Problems". Pearson/Prentice Hall.
5. Weisstein, Eric W.. "Maclaurin Series".
6. "Laurent series - Encyclopedia of Mathematics".
7. (26 January 2022). "Dirichlet series - Encyclopedia of Mathematics".
8. "Fourier series - Encyclopedia of Mathematics".
9. (2008). "Elementary Differential Equations with Boundary Value Problems". Pearson/Prentice Hall.
10. "DLMF: 5.11 Asymptotic Expansions".
11. Weisstein, Eric W.. "Exponential Function".
12. (5 June 2020). "Exponential function - Encyclopedia of Mathematics".