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.
This article is complete as far as it goes, but it could do with expansion. In particular: Historical note explaining Euler's involvement in a construct that had not at the time been defined -- Gauss had barely been born before Euler died You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 31$: Hypergeometric Functions: Miscellaneous Properties: $31.16$
- 1999: George E. Andrews, Richard Askey and Ranjan Roy: Special Functions: Chapter $\text {2}$. The Hypergeometric Functions
- 1935: W.N. Bailey: Generalized Hypergeometric Series Chapter $\text {1}$. The hypergeometric series
- Weisstein, Eric W. "Hypergeometric Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HypergeometricFunction.html