Definition:Gamma Function/Weierstrass Form
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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 known as
Some authors refer to the gamma function as Euler's gamma function, after Leonhard Paul Euler.
Some French sources call it the Eulerian function.
Also see
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 $24$
- 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