# integral representation of the hypergeometric function

When $\mathrm{\Re}c>\mathrm{\Re}b>0$, one has the representation

$$F(a,b;c;z)=\frac{\mathrm{\Gamma}(c)}{\mathrm{\Gamma}(b)\mathrm{\Gamma}(c-b)}{\int}_{0}^{1}{t}^{b-1}{(1-t)}^{c-b-1}{(1-tz)}^{-a}\mathit{d}t$$ |

Note that the conditions on $b$ and $c$ are necessary for the integral^{} to be convergent at the endpoints $0$ and $1$. To see that this integral indeed equals the hypergeometric function^{}, it suffices to consider the case $$ since both sides of the equation are analytic functions^{} of $z$. (This follows from the rigidity theorem for analytic functions although some care is required because the function^{} is multiply-valued.) With this assumption^{}, $$ if $t$ is a real number in the interval $[0,1]$ and hence, ${(1-tz)}^{-a}$ may be expanded in a power series^{}. Substituting this series in the right hand side of the formula^{} above gives

$$\frac{\mathrm{\Gamma}(c)}{\mathrm{\Gamma}(b)\mathrm{\Gamma}(c-b)}{\int}_{0}^{1}\sum _{k=0}^{\mathrm{\infty}}{t}^{b-1}{(1-t)}^{c-b-1}\frac{\mathrm{\Gamma}(k-a+1)}{\mathrm{\Gamma}(1-a)\mathrm{\Gamma}(k+1)}{(-tz)}^{k}dt$$ |

Since the series is uniformly convergent, it is permissible to integrate term-by-term. Interchanging integration and summation and pulling constants outside the integral sign, one obtains

$$\frac{\mathrm{\Gamma}(c)}{\mathrm{\Gamma}(b)\mathrm{\Gamma}(c-b)}\sum _{k=0}^{\mathrm{\infty}}\frac{\mathrm{\Gamma}(k-a+1)}{\mathrm{\Gamma}(1-a)\mathrm{\Gamma}(k+1)}{(-z)}^{k}{\int}_{0}^{1}{(1-t)}^{c-b-1}{t}^{b+k-1}\mathit{d}t$$ |

The integrals appearing inside the sum are Euler beta functions. Expressing them in terms of gamma functions^{} and simplifying, one sees that this integral indeed equals the hypergeometric function.

The hypergeometic function is multiply-valued. To obtain different branches of the hypergeometric function, one can vary the path of integration.

Title | integral representation of the hypergeometric function |
---|---|

Canonical name | IntegralRepresentationOfTheHypergeometricFunction |

Date of creation | 2013-03-22 14:35:14 |

Last modified on | 2013-03-22 14:35:14 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 33C05 |