Series expansion

File:Taylor cos.gif

An animation showing the cosine function being approximated by successive truncations of its Maclaurin series.

In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the partial sum of the omitted terms) can be described by an equation involving Big O notation (see also asymptotic expansion). The series expansion on an open interval will also be an approximation for non-analytic functions.<ref>Template:Cite book</ref>Template:Verify source

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.<ref name=":1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> More specifically, if a function <math>f: U\to\R</math> is infinitely differentiable around a point <math>x_0</math>, then the Taylor series of f around this point is given by

<math>\sum_{n=0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x - x_0)^n</math>

under the convention <math>0^0 := 1</math>.<ref name=":1" /><ref name=":2">Template:Cite book</ref> The Maclaurin series of f is its Taylor series about <math>x_0 = 0</math>.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name=":2" />

Laurent series

A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form <math display="inline">\sum_{k = -\infty}^{\infty} c_k (z - a)^k</math> and converges in an annulus.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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

File:ZetaSpiral.gif

Convergence and divergence of partial sums of the Dirichlet series defining the Riemann zeta function. Here, the yellow line represents the first fifty successive partial sums <math display="inline">\sum_{n = 1}^k n^{-s},</math> the magenta dotted line represents <math>\tfrac{n^{-s+1}}{-s+1} + \zeta(s),</math> and the green dot represents <math>\zeta(s)</math> as s is varied from -0.5 to 1.5.

A general Dirichlet series is a series of the form <math display="inline">\sum_{n = 1}^{\infty} a_ne^{-\lambda_n s}.</math> One important special case of this is the ordinary Dirichlet series <math display="inline">\sum_{n = 1}^{\infty}\frac{a_n}{n^s}.</math><ref name=":3">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Used in number theory.Template:Citation needed

Fourier series

A Fourier series is an expansion of periodic functions as a sum of many sine and cosine functions.<ref name=":4">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> More specifically, the Fourier series of a function <math>f(x)</math> of period <math>2L</math> is given by the expression<math display="block">a_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]</math>where the coefficients are given by the formulae<ref name=":4" /><ref>Template:Cite book</ref><math display="block">\begin{align} a_n &:= \frac{1}{L}\int_{-L}^L f(x)\cos\left(\frac{n\pi x}{L}\right)dx, \\ b_n &:= \frac{1}{L}\int_{-L}^L f(x)\sin\left(\frac{n\pi x}{L}\right)dx. \end{align}</math>

Other series

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

• Newtonian seriesTemplate:Citation needed

• 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.Template:Citation needed

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

File:Stirling series relative error.svg

The relative error in a truncated Stirling series vs. Template:Mvar, for 0 to 5 terms. The kinks in the curves represent points where the truncated series coincides with <math>\Gamma(n + 1).</math>

• The Stirling series<math display=block>\text{Ln}\Gamma\left(z\right)\sim\left(z-\tfrac{1}{2}\right)\ln z-z+\tfrac{1}{2}\ln\left(2\pi\right)+\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}</math>is an approximation of the log-gamma function.<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

Examples

The following is the Taylor series of <math>e^x</math>:<math display="block">e^x=\sum^{\infty}_{n=0}\frac{x^n}{n!}= 1 + x + \frac{x^2}{2} + \frac{x^3}{6}...</math><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

The Dirichlet series of the Riemann zeta function is<math display="block">\zeta(s) := \sum_{n = 1}^{\infty} \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \cdots</math><ref name=":3" />

References

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