Definition:Gamma Function/Weierstrass Form

Definition

The Weierstrass form of the gamma function is:

$\ds \frac 1 {\map \Gamma z} = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }$

where $\gamma$ is the Euler-Mascheroni constant.

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

Also see

• Equivalence of Definitions of Gamma Function

Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $16.12$: Other Definitions of the Gamma Function
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 38$: Infinite Products: $38.5$
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $24$