Euler's Integral Representation of Hypergeometric Function

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Theorem

Let $a, b, c \in \C$.

Let $\size x < 1$

Let $\map \Re c > \map \Re b > 0$.

Then:

$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{- a} \rd t$

in the $x$ plane cut along the real axis from $1$ to $\infty$.

where:

$\map \arg t = \map \arg {1 - t} = 0$
$\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$:
$\map F {a, b; c; x} := \ds \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} b^{\overline k} } {c^{\overline k} } \dfrac {x^k} {k!}$
$x^{\overline k}$ denotes the $k$th rising factorial power of $x$
$\map \Gamma {n + 1} = n!$ is the Gamma function.


Proof

Letting $\size x < 1$ and expanding the product of $\paren {1 - x t}^{-a}$:

\(\ds \paren {1 - x t}^{-a}\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \binom {-a} k \paren {-1}^k \paren {x t}^k\) Binomial Theorem - Complex Numbers
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \paren {\binom {a + k - 1} k \paren {-1}^k} \paren {-1}^k \paren {x t}^k\) Negated Upper Index of Binomial Coefficient
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \binom {a + k - 1} k \paren {x t}^k\) $\paren {-1}^{2k} = 1$
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {a + k - 1}!} {k! \paren {a - 1}! } x^k t^k\) Definition of Binomial Coefficient
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} x^k t^k\) Rising Factorial as Quotient of Factorials


Therefore:

\(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{-a} \rd t\) \(=\) \(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} x^k t^k \rd t\)
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} x^k \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} t^k \rd t\)
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} x^k \int_0^1 t^{k + b - 1} \paren {1 - t}^{c - b - 1} \rd t\) Product of Powers
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} x^k \dfrac {\map \Gamma {k + b} \map \Gamma {c - b} } {\map \Gamma {k + c} }\) Definition of Beta Function
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {x^k} {k!}\) Rising Factorial as Quotient of Factorials and $\map \Gamma {c - b}$ cancels
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \map F {a, b; c; x}\) Definition of Hypergeometric Function


Therefore:

$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c} {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{-a} \rd t$

$\blacksquare$


Also see


Source of Name

This entry was named for Leonhard Paul Euler.




Sources