Gamma function a generalized fictorial function

Question

I learned all the properties of gamma function. I did some of questions about gamma function. I know gamma function is generalized fictorial function that always positive. And i know that Gamma of 1/2 equals sqaure root pi. But i stuck on how to evaluate gamma 4/3. Please some one help me on this. Besides gamma function i knew about beta function which is also calles Euler beta function. I know the relation between Beta and Gamma funtion , Legendre's Dupocation formula and other topics related to Gamma and Beta function.

Answer 1

You can use

$$ \Gamma\left(\frac{4}{3}\right) = \Gamma\left(1 + \frac{1}{3}\right) = \frac{1}{3}\Gamma\left(\frac{1}{3}\right) $$

And that's pretty much it. $\Gamma(1/3)$ is a trascendental number, doesn't have a nice representation such as $\Gamma(1/2)$.

As for your statements, some of them are not true, e.g., $\Gamma(x)$ is not always positive, here's an example

$$ \Gamma\left(-\frac{1}{2}\right) = -2\sqrt{\pi} < 0 $$