Gaussian hypergeometric function

This article explains the “ordinary” hypergeometric function, also known as the hypergeometric function; the generalized hypergeometric function is explained under Generalized hypergeometric function .

The hypergeometric function , also referred to as Gaussian hypergeometric function or ordinary hypergeometric function , is understood in mathematics as a power series which is the solution of the hypergeometric differential equation . The function goes hand in hand with important mathematicians such as Leonhard Euler , Bernhard Riemann or Carl Friedrich Gauß . It is often used in mathematical physics. ${\displaystyle {}_{2}F_{1}(a,b;c;z)}$

definition

The hypergeometric function is defined by the power series

${\displaystyle {}_{2}F_{1}(a,b;c;z)=\sum _{k=0}^{\infty }{\frac {\Gamma (a+k)\,\Gamma (b+k)\,\Gamma (c)}{\Gamma (a)\,\Gamma (b)\,\Gamma (c+k)}}{\frac {z^{k}}{k!}}}$

for , where is not a non-positive integer and the function is the gamma function . ${\displaystyle a,b,c\in \mathbb {C} }$ ${\displaystyle c}$ ${\displaystyle \Gamma (\cdot )}$

convergence

This power series becomes a polynomial if or is a non-positive integer. ${\displaystyle a}$ ${\displaystyle b}$

Unless it is a polynomial, the power series for converges and is divergent for . Values ​​of the function for are determined by analytical continuation; Branch points are the points and . ${\displaystyle |z|<1}$ ${\displaystyle |z|>1}$ ${\displaystyle {}_{2}F_{1}(a,b;c;z)}$ ${\displaystyle |z|\geq 1,z\neq 1}$ ${\displaystyle 1}$ ${\displaystyle \infty }$

The following can be said about the convergence on the edge : The power series converges absolutely for if . If it holds and is real, the following convergence condition can be given: ${\displaystyle |z|=1}$ ${\displaystyle |z|=1}$ ${\displaystyle \operatorname {Re} \left(c-a-b\right)>0}$ ${\displaystyle \textstyle a+b\geq c}$ ${\displaystyle z}$

${\displaystyle \lim _{z\rightarrow 1}(1-z){\frac {\mathrm {d} \log(_{2}F_{1}(a,b;c;z^{2}))}{\mathrm {d} z}}=a+b-c}$.

The hypergeometric differential equation

→ Main article : Hypergeometric differential equation

As stated by Euler, the function satisfies a linear differential equation of the 2nd order . Substituting in shows that the series given above satisfies the hypergeometric differential equation below: ${\displaystyle w={}_{2}F_{1}(a,b;c;z)}$

${\displaystyle z(1-z){\frac {\mathrm {d} ^{2}w}{\mathrm {d} z^{2}}}+\left[c-(a+b+1)z\right]{\frac {\mathrm {d} w}{\mathrm {d} z}}-ab\,w=0}$

The series is therefore a particular solution of the differential equation. The solution applies to the area around the singular points and . Finally, with variants of the usual hypergeometric function, all solutions of the hypergeometric differential equation can be given. ${\displaystyle z=0,z=1}$ ${\displaystyle z=\infty }$

Euler also gave an integral representation for the solution of the hypergeometric differential equation:

${\displaystyle {}_{2}F_{1}(a,b;c;z)={\frac {\Gamma (c)}{\Gamma (b)\Gamma (c-b)}}\int _{0}^{1}t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,\mathrm {d} t}$

Every differential equation with three liftable singular points can be converted into the hypergeometric differential equation by transforming the variables.

Applications

Special functions

Many functions commonly used in mathematics can be expressed using the Gaussian hypergeometric function. Some identities that apply to are: ${\displaystyle |z|<1}$

${\displaystyle {}_{2}F_{1}\left(1,1;1;z\right)={\frac {1}{1-z}}}$

${\displaystyle {}_{2}F_{1}\left(-\alpha ,1,1;-z\right)=(1+z)^{\alpha },\qquad \alpha \in \mathbb {R} }$

${\displaystyle {}_{2}F_{1}\left(1,1;2;-z\right)={\frac {1}{z}}\ln(1+z)}$

${\displaystyle {}_{2}F_{1}\left({\frac {1}{2}},1;{\frac {3}{2}};z^{2}\right)={\frac {1}{2z}}\ln {\frac {1+z}{1-z}}}$

${\displaystyle {}_{2}F_{1}\left({\frac {1}{2}},{\frac {1}{2}};{\frac {3}{2}};z^{2}\right)={\frac {1}{z}}\arcsin z}$

${\displaystyle {}_{2}F_{1}\left({\frac {1}{2}},1;{\frac {3}{2}};-z^{2}\right)={\frac {1}{z}}\arctan z}$

Antiderivatives

With the hypergeometric function u. a. specify the following elementary antiderivatives :

${\displaystyle \int \cos ^{n}(x)\,\mathrm {d} x=-{\frac {\sqrt {\sin ^{2}(x)}}{\sin(x)}}\,{\frac {\cos ^{n+1}(x)}{n+1}}\,{}_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}},{\frac {n+3}{2}},\cos ^{2}(x)\right)}$

${\displaystyle \int \sin ^{n}(x)\,\mathrm {d} x=-\cos(x)\,{\frac {\sin ^{n+1}(x)}{\sin ^{2}(x)^{\frac {n+1}{2}}}}\,{}_{2}F_{1}\left({\frac {1}{2}},{\frac {1-n}{2}},{\frac {3}{2}},\cos ^{2}(x)\right)}$

Calculation of the hypergeometric function

The hypergeometric function can in principle be calculated using its series development. According to Gauss, the series converges for real as well as complex values . However, there are often unfavorable constellations that make the calculation much more difficult. The functional value in the range can practically already cause significant problems. Transformations and solutions for special function values ​​are helpful here. The following applies to the value : ${\displaystyle |z|<1}$ ${\displaystyle |z|>0,9}$ ${\displaystyle z=1}$

${\displaystyle {}_{2}F_{1}(a,b;c;1)={\frac {\Gamma (c)\,\Gamma (c-a-b)}{\Gamma (c-a)\,\Gamma (c-b)}}}$

Furthermore, the linear transformation

${\displaystyle {}_{2}F_{1}(a,b;c;z)=(1-z)^{c-a-b}{}_{2}F_{1}(c-a,c-b;c;z)}$

very helpful with unfavorable constellations of the coefficients. Further procedures, special solutions and transformations can be found on the web links given below .

See also

• Generalized hypergeometric function
• Gamma function

literature

• Leonhard Euler : Specimen transformationis singularis serierum . In: Nova Acta Academiae Scientarum Imperialis Petropolitinae . 12, 1801, pp. 58-70.

• Carl Friedrich Gauss : Disquisitiones generales circa seriem infinitam   ${\displaystyle 1+{\tfrac {a\cdot b}{1\cdot c}}\,x+{\tfrac {a(a+1)\cdot b(b+1)}{1\cdot 2\cdot c(c+1)}}\,x^{2}+...}$ . In: Commentationes recentiores Vol. II . , Göttingen 1813.

• Felix Klein : Lectures on the hypergeometric function , first part, first section, pp. 8–23, Springer, Berlin, reprint 1981.

• Ernst Eduard Kummer: About the hypergeometric series  ${\displaystyle 1+{\tfrac {\alpha \cdot \beta }{1\cdot \gamma }}~x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}x^{2}+{\tfrac {\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1\cdot 2\cdot 3\cdot \gamma (\gamma +1)(\gamma +2)}}x^{3}+{\mbox{etc.}}}$ . In: Journal for pure and applied mathematics . 15, 1836.

• Bernhard Riemann: Contributions to the theory of Gaussian by the number F (α, β, γ, x) representable functions . In: Verlag der Dieterichschen Buchhandlung (Hrsg.): Treatises of the Mathematical Class of the Royal Society of Sciences in Göttingen . 7, Göttingen, 1857.

• Arthur Erdélyi , Wilhelm Magnus, Fritz Oberhettinger, Francesco G. Tricomi: Higher transcendental functions, Volume I, Chapter II, pages 56–99, New York - Toronto - London, McGraw – Hill Book Company, Inc., 1953, ISBN 978- 0-89874-206-0 , pdf

Web links

• Transformations of Variable (collection of linear, quadratic and cubic transformations from NIST)
• Gauss Hypergeometric Function (List of Identities from Wolfram Research)
• John Pearson, Computation of Hypergeometric Functions ( University of Oxford , MSc Thesis)

Individual evidence

1. ↑ J. Quigley, KJ Wilson, L. Walls, T. Bedford: A Bayes linear Bayes Method for Estimation of Correlated Event Rates In: Risk Analysis 2013 doi = 10.1111 / risa.12035