Series expansion explained

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).[1]

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.[2]

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.[3] More specifically, if a function

f:U\to\R

is infinitely differentiable around a point

x0

, then the Taylor series of f around this point is given by
infty
\sum
n=0
f(n)(x0)
n!

(x-

n
x
0)

under the convention

00:=1

.[4] The Maclaurin series of f is its Taylor series about

x0=0

.[5]

Laurent series

A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form \sum_^ c_k (z - a)^k and converges in an annulus.[6] 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_^ a_ne^. One important special case of this is the ordinary Dirichlet series \sum_^\frac.[7] Used in number theory.

Fourier series

A Fourier series is an expansion of periodic functions as a sum of many sine and cosine functions.[8] More specifically, the Fourier series of a function

f(t)

of period

2L

is given by the expressiona_0 + \sum_^ \left[a_n\cos\left(\frac{n\pi t}{L}\right) + b_n\sin\left(\frac{n\pi t}{L}\right)\right]where the coefficients are given by the formulae[9] \begina_n &:= \frac\int_^L f(t)\cos\left(\frac\right)dt, \\b_n &:= \frac\int_^L f(t)\sin\left(\frac\right)dt.\end

Other series

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

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

Examples

The following is the Taylor series of

ex

:e^x=\sum^_\frac= 1 + x + \frac + \frac...[11] [12]

The Dirichlet series of the Riemann zeta function is\zeta(s) := \sum_^ \frac = \frac + \frac + \cdots

References

  1. Web site: 2013-11-07. Series and Expansions. 2021-12-24. Mathematics LibreTexts. en.
  2. Book: Gil. Amparo. Numerical Methods for Special Functions. Segura. Javier. Temme. Nico M.. 2007-01-01. SIAM. 978-0-89871-782-2. en.
  3. Web site: 27 December 2013 . Taylor series - Encyclopedia of Mathematics . 22 March 2022 . encyclopediaofmath.org.
  4. Book: Edwards . C. Henry . Elementary Differential Equations with Boundary Value Problems . Penney . David E. . 2008 . 978-0-13-600613-8 . 196 . Pearson/Prentice Hall . en.
  5. Web site: Weisstein . Eric W. . Maclaurin Series . 2022-03-22 . mathworld.wolfram.com . en.
  6. Web site: Laurent series - Encyclopedia of Mathematics . 2022-03-22 . encyclopediaofmath.org.
  7. Web site: 26 January 2022 . Dirichlet series - Encyclopedia of Mathematics . 22 March 2022 . encyclopediaofmath.org.
  8. Web site: Fourier series - Encyclopedia of Mathematics . 2022-03-22 . encyclopediaofmath.org.
  9. Book: Edwards . C. Henry . Elementary Differential Equations with Boundary Value Problems . Penney . David E. . 2008 . 978-0-13-600613-8 . 558, 564 . Pearson/Prentice Hall . en.
  10. Web site: DLMF: 5.11 Asymptotic Expansions . 22 March 2022 . dlmf.nist.gov.
  11. Web site: Weisstein. Eric W.. Exponential Function. 2021-08-12. mathworld.wolfram.com. en.
  12. Web site: 5 June 2020. Exponential function - Encyclopedia of Mathematics. 12 August 2021. encyclopediaofmath.org.