Gamma Function of Minus 5 over 2

Theorem

$\map \Gamma {-\dfrac 5 2} = -\dfrac {8 \sqrt \pi} {15}$

where $\Gamma$ denotes the Gamma function.


Proof

\(\ds \map \Gamma {-\dfrac 3 2}\) \(=\) \(\ds -\dfrac 5 2 \map \Gamma {-\dfrac 5 2}\) Gamma Difference Equation
\(\ds \leadsto \ \ \) \(\ds \map \Gamma {-\dfrac 5 2}\) \(=\) \(\ds -\dfrac 2 5 \map \Gamma {-\dfrac 3 2}\)
\(\ds \) \(=\) \(\ds -\dfrac 2 5 \paren {\dfrac {4 \sqrt \pi} 3}\) Gamma Function of Minus 3 over 2
\(\ds \) \(=\) \(\ds -\dfrac {8 \sqrt \pi} {15}\)

$\blacksquare$


Sources

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