# Gamma function - why is special ?

• Mar 1st 2010, 02:06 AM
pawel
Gamma function - why is special ?
How to prove that gamma function is a special function ? In case of eg. expotetential integral one can reduce the problem to a linear differential equation and can ask for solvability of this equation. In this case we can use the differential algebra machinery to prove its unsolvability. But can we prove that gamma function is a special function by similar way ? As I know this function does not satisfy any simply differential equation.
• Mar 2nd 2010, 01:48 PM
wonderboy1953
Question
What do you mean by a special function?
• Mar 3rd 2010, 01:26 AM
pawel
Gamma function
A special function=a non-elementary function in my problem. A non elementary function can't be built from a finite number of algebraic and expotentials, logarithms, sine, cosine etc.
• Mar 3rd 2010, 03:15 AM
CaptainBlack
Quote:

Originally Posted by pawel
A special function=a non-elementary function in my problem. A non elementary function can't be built from a finite number of algebraic and expotentials, logarithms, sine, cosine etc.

It is an integral of elementary functions (and I presume the Risch algorithm can be used to prove that this integral cannot be expressed in closed form in terms of elemenatry functions).

CB
• Mar 3rd 2010, 06:17 AM
pawel
I cant understand. The gamma function is integral of some elementary function ? If yes, thus its derivative should be an elemetary function. As I know it isnt true
• Mar 3rd 2010, 07:41 AM
chisigma
The definition of the 'Gamma Function' is...

$\displaystyle \Gamma (z) = \int_{0}^{\infty} t^{z-1}\cdot e^{-t}\cdot dt$ , $\displaystyle \Re (z) > 0$ (1)

... so that defining it as 'primitive of an elementary trascendent function' is erroneous. The [complex] derivative of (1) is...

$\displaystyle \Gamma^{'} (z) = \int_{0}^{\infty} t^{z-1}\cdot \ln t \cdot e^{-t}\cdot dt$ , $\displaystyle \Re (z) >0$ (2)

... which of course in not an elementary trascendent function. The reason for which the Gamma Function is 'special' is that it represents the first historical example of analytic extension of a complex function, a powerful kay that opens many 'doors' of the mathematical world ...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$