# Two Questions about Gamma Function Terminology

Gamma function is also known as generalized factorial function .

1. Why does the term "generalized" have been used?

2. Why is the Gamma function called Euler's second integral?

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generalized means that the gamma functions extends the factorial funcion ($\Gamma(n) = (n-1)!$, $n \in \mathbb{N}$)... this is the kind of questions that can be answered perfectly by wikipedia: en.wikipedia.org/wiki/Gamma_function – user67133 May 8 '13 at 9:46

The Factorial is defined only for integers: $$n!=n(n-1)(n-2)\cdots3\cdot2\cdot1$$ The Gamma function, $\Gamma(x)$, is defined for general complex arguments $$\Gamma(z)=\int_0^\infty t^{z-1}\,e^{-t}\,\mathrm{d}t$$ where the integral converges if $\mathrm{Re}(z)\gt0$, but $\Gamma(x)$ can be analytically continued to the whole complex plane minus some isolated points using the reflection formula: $$\Gamma(z)\Gamma(1-z)=\frac\pi{\sin(\pi z)}$$
$\Gamma(x)$ is called the generalized factorial since $n!=\Gamma(n+1)$.
No, $z$ is often used to indicate a complex variable and $x$ a real variable. – robjohn May 9 '13 at 9:45