Gamma Function of Minus 3 over 2

Theorem

$\map \Gamma {-\dfrac 3 2} = \dfrac {4 \sqrt \pi} 3$

where $\Gamma$ denotes the Gamma function.


Proof

\(\ds \map \Gamma {-\dfrac 1 2}\) \(=\) \(\ds -\dfrac 3 2 \map \Gamma {-\dfrac 3 2}\) Gamma Difference Equation
\(\ds \leadsto \ \ \) \(\ds \map \Gamma {-\dfrac 3 2}\) \(=\) \(\ds -\dfrac 2 3 \map \Gamma {-\dfrac 1 2}\)
\(\ds \) \(=\) \(\ds -\dfrac 2 3 \paren {-2 \sqrt \pi}\) Gamma Function of Minus One Half
\(\ds \) \(=\) \(\ds \dfrac {4 \sqrt \pi} 3\)

$\blacksquare$


Sources

  • 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Gamma Function: $33 \ \text{(b)}$
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