Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Gamma function is also known as generalized factorial function .

why does the term "generalized" have been used ?

Again, why is the Gamma function called Euler's second integral?

share|improve this question
3  
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 –  user01123581321345589144... May 8 '13 at 9:46

1 Answer 1

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.

share|improve this answer
    
thank you very much –  harry 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) ? –  harry 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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.