Gamma Function of One Half
Theorem
Let $\Gamma$ denote the Gamma function.
Then:
- $\map \Gamma {\dfrac 1 2} = \sqrt \pi$
Decimal Expansion
The decimal expansion of $\map \Gamma {\dfrac 1 2}$ starts:
- $\map \Gamma {\dfrac 1 2} = 1 \cdotp 77245 \, 38509 \, 05516 \, 02729 \, 81674 \, 83341 \, 14518 \, 27975 \ldots$
Proof 1
From the definition of the Beta function:
- $\map \Beta {x, y} := \dfrac {\map \Gamma x \map \Gamma y} {\map \Gamma {x + y} }$
Setting $x = y = \dfrac 1 2$:
| \(\ds \map \Beta {\dfrac 1 2, \dfrac 1 2}\) | \(=\) | \(\ds \frac {\map \Gamma {\dfrac 1 2} \map \Gamma {\dfrac 1 2} } {\map \Gamma {\dfrac 1 2 + \dfrac 1 2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\map \Gamma {\dfrac 1 2} }^2\) |
Then from Beta Function of Half with Half:
- $\map \Beta {\dfrac 1 2, \dfrac 1 2} = \pi$
Hence the result.
$\blacksquare$
Proof 2
From Euler's Reflection Formula:
- $\forall z \notin \Z: \map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$
Setting $z = \dfrac 1 2$:
| \(\ds \map \Gamma {\frac 1 2} \map \Gamma {\frac 1 2}\) | \(=\) | \(\ds \frac \pi {\map \sin {\frac \pi 2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac \pi 1\) | Sine of Right Angle | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \Gamma {\frac 1 2}\) | \(=\) | \(\ds \pm \sqrt \pi\) |
By definition of the gamma function:
- $\forall z \in \R_{\ge 0}: \map \Gamma z > 0$
and so the negative square root can be discarded.
Hence:
- $\map \Gamma {\dfrac 1 2} = \sqrt \pi$
as required.
$\blacksquare$
Proof 3
| \(\ds \map \Gamma {\dfrac 1 2}\) | \(=\) | \(\ds \int_0^{\to \infty} t^{-\frac 1 2} e^{-t} \rd t\) | Definition of Gamma Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^{\to \infty} u^{-1} e^{-u^2} 2 u \rd u\) | Integration by Substitution, $\map \phi u = u^2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \int_0^{\to \infty} e^{-u^2} \rd u\) | Linear Combination of Definite Integrals | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_{\to -\infty}^{\to \infty} e^{-u^2} \rd u\) | Definite Integral of Even Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt \pi\) | Gaussian Integral |
$\blacksquare$
Proof 4
| \(\ds \map \Gamma {\frac 1 2}\) | \(=\) | \(\ds \frac {0!} {2^0 0!} \sqrt \pi\) | 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): $\S 16$: Special Values for the Gamma Function: $16.5$
- 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): $1 \cdotp 77245 \, 38509 \, 05516 \, 02729 \, 81674 \, 83341 \, 14518 \, 27975 \ldots$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): gamma function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): gamma function
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 1$: Special Constants: $1.6.$
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 25$: The Gamma Function: Special Values for the Gamma Function: $25.5.$