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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: – user67133 May 8 '13 at 9:46
up vote 6 down vote accepted

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 $n!=\Gamma(n+1)$.

The Beta function and the Gamma function are called Euler's first and second integrals.

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thank you very much – ABC May 9 '13 at 6:10
You're welcome. Was anything left unanswered? – robjohn May 9 '13 at 7:10
is there difference between Γ(z) & Γ(x) ? – ABC May 9 '13 at 9:17
No, $z$ is often used to indicate a complex variable and $x$ a real variable. – robjohn May 9 '13 at 9:45

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