Euler's integral representation of the hypergeometric function

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 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.

Going deeper

For a student

This representation is the main tool for proving identities for ${}_2F_1$. Setting $z = 1$ gives Gauss's hypergeometric theorem by the beta integral, and the substitution $t \mapsto 1 - t$ gives the Pfaff transformation. 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.