Gamma Function of One Half
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Theorem
Let $\Gamma$ denote the Gamma function.
Then:
- $\map \Gamma {\dfrac 1 2} = \sqrt \pi$
Its decimal expansion starts:
- $\map \Gamma {\dfrac 1 2} = 1 \cdotp 77245 \, 38509 \, 05516 \, 02729 \, 81674 \, 83341 \, 14518 \, 27975 \ldots$
This sequence is A002161 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof 1
From the definition of the Beta function:
- $\Beta \left({x, y}\right) := \dfrac {\Gamma \left({x}\right) \Gamma \left({y}\right)} {\Gamma \left({x + y}\right)}$
Setting $x = y = \dfrac 1 2$:
\(\ds \Beta \left({\dfrac 1 2, \dfrac 1 2}\right)\) | \(=\) | \(\ds \frac {\Gamma \left({\dfrac 1 2}\right) \Gamma \left({\dfrac 1 2}\right)} {\Gamma \left({\dfrac 1 2 + \dfrac 1 2}\right)}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \left({\Gamma \left({\dfrac 1 2}\right)}\right)^2\) |
Then from Beta Function of Half with Half:
- $\Beta \left({\dfrac 1 2, \dfrac 1 2}\right) = \pi$
Hence the result.
$\blacksquare$
Proof 2
From Euler's Reflection Formula:
- $\forall z \notin \Z: \Gamma \left({z}\right) \Gamma \left({1 - z}\right) = \dfrac \pi {\sin \left({\pi z}\right)}$
Setting $z = \dfrac 1 2$:
\(\ds \Gamma \left({\frac 1 2}\right) \Gamma \left({\frac 1 2}\right)\) | \(=\) | \(\ds \frac \pi {\sin \left({\frac \pi 2}\right)}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 1\) | Sine of Right Angle | |||||||||||
\(\ds \implies \ \ \) | \(\ds \Gamma \left({\frac 1 2}\right)\) | \(=\) | \(\ds \pm \sqrt \pi\) |
By definition of the $\Gamma$ function:
- $\forall z \in \R_{\ge 0}: \Gamma \left({z}\right) > 0$
and so the negative square root can be discarded.
Hence:
- $\Gamma \left({\dfrac 1 2}\right) = \sqrt \pi$
as required.
$\blacksquare$
Proof 3
\(\ds \Gamma \left({\dfrac 1 2}\right)\) | \(=\) | \(\ds \int_0^{\to \infty} t^{-\frac 1 2} e^{-t} \ \mathrm d t\) | Definition of Gamma Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^{\to \infty} u^{-1} e^{-u^2} 2 u \ \mathrm d u\) | Integration by Substitution, $\phi \left({u}\right) = u^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \int_0^{\to \infty} e^{-u^2} \ \mathrm d u\) | Linear Combination of Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_{\to -\infty}^{\to \infty} e^{-u^2} \ \mathrm d u\) | Definite Integral of Even Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\pi}\) | Gaussian Integral |
$\blacksquare$
Proof 4
\(\ds \Gamma \left({\frac 1 2}\right)\) | \(=\) | \(\ds \frac {0!} {2^0 0!} \sqrt \pi\) | Gamma Function of Positive Half-Integer | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt \pi\) | Factorial of Zero |
$\blacksquare$
Proof 5
\(\ds \map \Gamma 1 \, \map \Gamma {\frac 1 2}\) | \(=\) | \(\ds 2^0 \sqrt \pi \ \map \Gamma 1\) | Legendre's Duplication Formula | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 0! \, \map \Gamma {\frac 1 2}\) | \(=\) | \(\ds 0! \, \sqrt \pi\) | Definition of Gamma Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \Gamma {\frac 1 2}\) | \(=\) | \(\ds \sqrt \pi\) | Factorial of Zero |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {I}$. The Gamma function: $2$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 1$: Special Constants: $1.23$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $16.5$: Special Values for the Gamma Function
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 772 \, 453 \, 850 \, 905 \, 516 \, 027 \, 298 \, 167 \, 483 \, 341 \, 145 \, 182 \, 797 \ldots$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: Exercise $9$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 77245 \, 38509 \, 05516 \, 02729 \, 81674 \, 83341 \, 14518 \, 27975 \ldots$