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...

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...

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...

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

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

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...

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

mechanics, the cluster expansion (also called the high temperature expansion or hopping expansion) is a power series expansion of the partition function...

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...

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)...

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...

{\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...

{\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...

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...

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

}}-{\frac {\pi ^{6}x^{6}}{7!}}+\cdots } Euler famously compared this series to the expansion of the infinite product form to solve the Basel problem. The product...

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

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

order by order by a series expansion in the number of virtual particles participating in the interaction. Every term in the expansion may be understood...

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...

Fourier series is the expansion of a square integrable function into a sum of square integrable orthogonal basis functions. The standard Fourier series uses...

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

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

into an asymptotic series. To obtain a series expansion of the inverse gamma function one can first compute the series expansion of the reciprocal gamma...

Evaluations of the digamma function at rational values. The Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes...

In stochastic processes, Kramers–Moyal expansion refers to a Taylor series expansion of the master equation, named after Hans Kramers and José Enrique...

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

_{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...