Definition:Gamma Function
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Definition
Standard Definition
The Gamma function $\Gamma: \C \to \C \ $ is defined, for the open right half-plane, as:
- $\displaystyle \Gamma \left({z}\right) = \int_0^{\to \infty} t^{z-1} e^{-t} \ \mathrm d t$
and for all other values of $z$ except the non-positive integers as:
- $\Gamma \left({z + 1}\right) = z \Gamma \left({z}\right)$
Other equivalent definitions exist, as follows.
Weierstrass Form
Of note is the Weierstrass form:
- $\displaystyle \frac 1 {\Gamma \left({z}\right)} = z e^{\gamma z} \prod_{n=1}^\infty \left({\left({ 1 + \frac z n}\right) e^{\frac {-z} n}}\right)$
where $\gamma$ is the Euler-Mascheroni constant. The Weierstrass expression is valid for all $\C$.
Euler Form
Another important form of the Gamma function is the Euler form:
- $\displaystyle \Gamma \left({z}\right) = \frac 1 z \prod_{n=1}^\infty \left({ \left({1 + \frac 1 n}\right)^z \left({1 + \frac z n}\right)^{-1}}\right) = \lim_{m \to \infty} \frac {m^z m!} {z \left({z+1}\right) \left({z+2}\right) \ldots \left({z+m}\right)}$
which is valid except for $z \in \left\{{0, -1, -2, \ldots}\right\} \ $.
Extension of Factorial
The Gamma function can be seen to be an extension to the complex plane of the factorial:
- $\displaystyle n! = \Gamma \left({n+1}\right) = n \Gamma \left({n}\right)$
Hence we have:
- $\displaystyle n! = \lim_{m \to \infty} \frac {m^n m!} {\left({n+1}\right) \left({n+2}\right) \ldots \left({n+m}\right)}$
See Also
- Equivalence of Gamma Function Definitions
- Zeroes of the Gamma Function
- Poles of the Gamma Function
- Gamma Difference Equation
Historical Note
The symbol $\Gamma \left({x}\right)$ was introduced by Adrien-Marie Legendre.
Sources
- Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.5$
- Murray R. Spiegel: Mathematical Handbook of Formulas and Tables (1968): $16.1, \ 16.11, \ 16.12$
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous): $\S 17.3$
