Properties of Gamma(ln(x))?

I'm working with some equations which have forced me to deal with the gamma function, which is pretty much way over my head.

Do any neat shortcuts or identities arise when the argument of the gamma function is logged? As in $\displaystyle \Gamma(y\ln{x})$

Thanks

Re: Properties of Gamma(ln(x))?

Quote:

Originally Posted by

**rainer** I'm working with some equations which have forced me to deal with the gamma function, which is pretty much way over my head.

Do any neat shortcuts or identities arise when the argument of the gamma function is logged? As in $\displaystyle \Gamma(y\ln{x})$

Thanks

Many of the properties of the gamma function are tabulated in Abramowitz and Stegun. This can be found online (and downloaded), one link is:

Abramowitz and Stegun: Handbook of Mathematical Functions

CB

Re: Properties of Gamma(ln(x))?

Quote:

Originally Posted by

**rainer** I'm working with some equations which have forced me to deal with the gamma function, which is pretty much way over my head.

Do any neat shortcuts or identities arise when the argument of the gamma function is logged? As in $\displaystyle \Gamma(y\ln{x})$

Thanks

An expression of $\displaystyle \frac{1}{\Gamma(u)}$ as integral in $\displaystyle \mathbb{C}$ can be derived from thwe well known inverse Laplace Transform...

$\displaystyle \frac{1}{s^{u}}= \mathcal{L}^{-1} \{\frac{t^{u-1}}{\Gamma(u)}\}\ ,\ u>0 $ (1)

Using the Bromwitch integration formula and setting $\displaystyle t=1$ we have...

$\displaystyle \frac{1}{\Gamma(u)}= \frac{1}{2\ \pi\ i}\ \int_{c-i \infty}^{c+i \infty} \frac{e^{s}}{s^{u}}\ ds$ (2)

... where c>0 is arbitrary. Now You can set in (2) $\displaystyle u= \ln x$ and observe 'what happens'... may be there is a way to solve the integral...

Kind regards

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