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...ate.net/publication/322252136 |title=Generalized incomplete gamma function and its application |date=2018-01-14 |accessdate=2020-01-08}}</ref><ref>{{Cite :<math>\gamma(\alpha,x;b)=\int_0^xt^{\alpha-1}e^{-t-\frac{b}{t}}~dt</math> ...

and is closely related to [[Bessel function]]s. ...=Heinrich Friedrich Weber|first=H. F.|last=Weber|year=1879}}, is a closely related function defined by ...

...3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New Yo ...the [[q-Hahn polynomials]] ''Q''_''n''(''x'';α,β, ''N'';''q''), and so on. ...

In [[mathematics]], the '''Struve functions''' {{math|'''H'''_''α''(''x'')}}, are solutions {{math|''y''(''x' ...lpha^2 \right )y = \frac{4\left (\frac{x}{2}\right)^{\alpha+1}}{\sqrt{\pi}\Gamma \left (\alpha+\frac{1}{2} \right )}</math> ...

...inate surfaces]] of parabolic cylindrical coordinates. Parabolic cylinder functions occur when [[separation of variables]] is used on [[Laplace equation|Laplac In [[mathematics]], the '''parabolic cylinder functions''' are [[special function]]s defined as solutions to the differential equat ...

...when the Weierstrass invariants satisfy {{math|''g''₂ {{=}} 1}} and {{math|''g''₃ {{=}} 0}}. ...mniscatic case, the minimal half period {{math|''ω''₁}} is real and equal to ...

...s a [[special function]] that is closely related to the [[gamma function]] and to [[binomial coefficient]]s. It is defined by the [[integral]] ...studied by [[Leonhard Euler|Euler]] and [[Adrien-Marie Legendre|Legendre]] and was given its name by [[Jacques Philippe Marie Binet|Jacques Binet]]; its s ...

...als, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia ar ...noted n), and μ=0 are the Legendre polynomials {{math|''P_n''}}; and when ...

...ole [[complex plane]], and is then called a '''Dirichlet ''L''-function''' and also denoted ''L''(''s'', χ). These functions are named after [[Peter Gustav Lejeune Dirichlet]] who introduced them in { ...

...'''31''' (1900), 264–314.</ref> It can be written in terms of the [[double gamma function]]. :<math> G(1+z)=(2\pi)^{z/2} \exp\left(- \frac{z+z^2(1+\gamma)}{2} \right) \, \prod_{k=1}^\infty \left\{ \left(1+\frac{z}{k}\right)^k \ex ...

...le integrals of trigonometric functions|List of integrals of trigonometric functions}} [[Image:sine cosine integral.svg|right|thumb|Si(x) (blue) and Ci(x) (green) plotted on the same plot.]] ...

...ecial function|the Airy stress function employed in solid mechanics|Stress functions}} ...onomer [[George Biddell Airy]] (1801–1892). The function Ai(''x'') and the related function '''Bi(''x'')''', are linearly independent solutions to the [[diffe ...

...ing Hahn polynomials, [[Meixner polynomials]], [[Krawtchouk polynomials]], and [[Charlier polynomials]]. Sometimes the Hahn class is taken to include [[li ...3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New Yo ...

...'x'') is a [[special function]]. It is relevant in problems of [[physics]] and has [[number theory|number theoretic]] significance. In particular, accordi ...}} has a [[mathematical singularity|singularity]] at {{math|1=''t'' = 1}}, and the integral for {{math|''x'' > 1}} is interpreted as a [[Cauchy principal ...

...' or '''Dawson integral'''<ref>{{dlmf|id=7|title=Error Functions, Dawson's and Fresnel Integrals|first=N. M. |last=Temme}}</ref> It is closely related to the [[error function]] erf, as ...

...ther". There are several common standard forms of confluent hypergeometric functions: ...confluent hypergeometric function of the first kind. There is a different and unrelated [[Kummer's function]] bearing the same name. ...

...''q''-analogue]] generalizations of [[generalized hypergeometric series]], and are in turn generalized by [[elliptic hypergeometric series]]. ...ypergeometric series, the '''unilateral basic hypergeometric series''' φ, and the more general '''bilateral basic hypergeometric series''' ψ. ...

...or [[complex number|complex]] arguments ''s'' with Re(''s'')&nbsp;>&nbsp;1 and ''q'' with Re(''q'')&nbsp;>&nbsp;0 by ...eries is [[absolutely convergent]] for the given values of ''s'' and ''q'' and can be extended to a [[meromorphic function]] defined for all ''s''&ne;1. T ...

...after the Czech mathematician [[Mathias Lerch]] [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lerch.html]. A related function, the '''Lerch transcendent''', is given by ...

...thematics)|function]]s are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variab ...n#The_hypergeometric_series|Pochhammer symbol]]. For other values of ''x'' and ''y'' the function ''F''₁ can be defined by [[analytic continuat ...