---
tag: 812THX456R
kind: statement
role: theorem
name: Euler's integral representation of the hypergeometric function
msc: ["33C05", "33B15"]
uses: [463X5V31D9, 0H5SYBSED9, QHNWVSC6AQ]
about: [463X5V31D9]
sources:
  - url: https://en.wikipedia.org/wiki/Hypergeometric_function
---

## Statement

Let $a, b, c \in \C$ with $\operatorname{Re} c > \operatorname{Re} b > 0$, and let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
Then for every $z \in \C \setminus [1, \infty)$,
$${}_2F_1(a, b; c; z) = \frac{\Gamma(c)}{\Gamma(b) \, \Gamma(c - b)} \int_0^1 t^{b-1} (1 - t)^{c - b - 1} (1 - zt)^{-a} \, dt,$$
where ${}_2F_1$ is the [[hypergeometric function|463X5V31D9]] and $(1 - zt)^{-a}$ is the principal power.

## Remarks

For $\abs{z} < 1$, expand $(1 - zt)^{-a} = \sum_n \frac{(a)_n}{n!} z^n t^n$ by the binomial series and integrate term by term.
The $n$-th term gives the [[beta function|QHNWVSC6AQ]] value $B(b + n, c - b) = \frac{\Gamma(b + n) \Gamma(c - b)}{\Gamma(c + n)}$, and $\Gamma(b + n)/\Gamma(b) = (b)_n$, $\Gamma(c + n)/\Gamma(c) = (c)_n$, which gives the series of ${}_2F_1$.
The integral is holomorphic in $z \in \C \setminus [1, \infty)$, because $1 - zt$ never lies on $(-\infty, 0]$ there, so the identity extends by analytic continuation.

Example: with $a = 1$, $b = 1$, $c = 2$ the integral is $\int_0^1 \frac{dt}{1 - zt} = -\frac{\log(1 - z)}{z}$, which is ${}_2F_1(1, 1; 2; z)$.

The hypothesis $\operatorname{Re} c > \operatorname{Re} b > 0$ is needed for convergence of the integral at $t = 0$ and $t = 1$.
For $b = c$ the factor $(1 - t)^{-1}$ makes the integral diverge at $t = 1$, although ${}_2F_1(a, b; b; z) = (1 - z)^{-a}$ is perfectly defined.
A Pochhammer contour integral removes this restriction.

## Explanations

### L1 human

The hypergeometric function is first defined by a power series that only converges for $\abs{z} < 1$.
Euler's formula writes it as an integral over $[0, 1]$ of an explicit function.
The integral makes sense for every $z$ off the cut $[1, \infty)$, so it shows directly how to extend the function there.

### L2 human

This representation is the main tool for proving identities for ${}_2F_1$.
Setting $z = 1$ gives [[Gauss's hypergeometric theorem|1FC0T0GCCX]] by the beta integral, and the substitution $t \mapsto 1 - t$ gives the [[Pfaff transformation|KAAJ0JJ2GF]].
It also exhibits ${}_2F_1$ as a period integral of a twisted differential form, which leads to the monodromy and to generalizations such as Appell and Lauricella functions.