Definition:Gamma Function/Hankel Form

Definition

The Hankel form of the gamma function is:

$\ds \frac 1 {\map \Gamma z} = \dfrac 1 {2 \pi i} \oint_\HH \frac {e^t \rd t} {t^z}$

where $\HH$ is the contour starting at $-\infty$, circling the origin in an anticlockwise direction, and returning to $-\infty$.

The Hankel form is valid for all $\C$.

Also see

• Equivalence of Definitions of Gamma Function

Source of Name

This entry was named for Hermann Hankel.

Sources

• 1864: Hermann Hankel: Die Euler'schen Integrale bei unbeschränkter Variabilität des Argumentes (Z. Math. Phys. Vol. 9: pp. 1 – 21)

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: $(17)$