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...

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives...

express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse...

A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system...

c| = r such that no analytic continuation of the series can be defined at x. The power series expansion of the inverse function of an analytic function...

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property...

The expansion of the universe is the increase in distance between gravitationally unbound parts of the observable universe with time. It is an intrinsic...

defined by the following power series expansions. These series are also known as the Taylor series or Maclaurin series of these trigonometric functions:...

A Fourier series (/ˈfʊrieɪ, -iər/) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a...

theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. Lagrange inversion...

virial expansion is a model of thermodynamic equations of state. It expresses the pressure P of a gas in local equilibrium as a power series of the density...

{\displaystyle n} -th Bell polynomial, we have in particular the Laurent series expansion of the gamma function Γ ( z ) = 1 z − γ + 1 2 ( γ 2 + π 2 6 ) z − 1...

calculated by finding Laurent series expansions, and one can define the residue as the coefficient a−1 of a Laurent series. The concept can be used to provide...

television series that began airing on The CW on October 3, 2013. It is a spin-off of The Vampire Diaries and the first television series expansion of the...

explain the so-called "compactness" problem: absent this ultra-relativistic expansion, the ejecta would be optically thick to pair production at typical peak...

{\displaystyle \gamma } is the Euler–Mascheroni constant. It has the series expansion Chi ⁡ ( x ) = γ + ln ⁡ ( x ) + x 2 4 + x 4 96 + x 6 4320 + x 8 322560...

binomial expansion is an increasing function of n, it follows from the monotone convergence theorem for series that the sum of this infinite series is equal...

Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written...

asymptotic expansion of Γ ( 1 / x ) {\displaystyle \Gamma (1/x)} . Evaluations of the digamma function at rational values. The Laurent series expansion for the...

analog of the BCH expansion", as outlined in a subsequent section. The equation above constitutes the Magnus expansion, or Magnus series, for the solution...

In mathematics, Fourier–Bessel series is a particular kind of generalized Fourier series (an infinite series expansion on a finite interval) based on...

_{q=0}^{N-1}e^{ix\sin {(2q+1)\pi /N}}e^{-i(2q+1)\pi p/N}} A series expansion using Bessel functions (Kapteyn series) is 1 1 − z = 1 + 2 ∑ n = 1 ∞ J n ( n z ) . {\displaystyle...

Res(f, c) of f at c is the coefficient a−1 of (z − c)−1 in the Laurent series expansion of f around c. Various methods exist for calculating this value, and...

convergent series expansion, adding series that converge in the same region. If all series expansions are divergent series or null series (all series terms zero)...

The Bantu expansion was a major series of migrations of the original Proto-Bantu-speaking group, which spread from an original nucleus around West-Central...

and limits. It can derive closed-form series expansions as well as terms of Taylor-Maclaurin-Laurent series. It can perform matrix manipulations with...

(sufficiently small) concrete values of x, and which has the formal series as its series expansion; this explains the designation "generating functions". However...

results as long as the expansion parameter, say α, is very small. Typically, the results are expressed in terms of finite power series in α that seem to converge...

Seventeen expansion packs have been released for the 2014 life simulation video game The Sims 4, the fourth major title in The Sims series. All expansion packs...

sum is called a Chebyshev series or a Chebyshev expansion. Since a Chebyshev series is related to a Fourier cosine series through a change of variables...