Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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5
votes
2answers
876 views

Two ways of defining the gamma function $\Gamma (x)$. How to show they are equivalent?

In the Portuguese book Análise Matemática (Mathematical Analysis) by C. Sarrico, it is proved there exist $$\displaystyle\lim_{n\rightarrow +\infty}f_{n}(x)=\lim_{n\rightarrow +\infty }e^{\log ...
7
votes
1answer
240 views

Is the Gamma function superadditive?

A function $f$ is superadditive if $f(x) + f(y) \le f(x+y)$. The question is: Does a real number $a$ exists such that for all real numbers with $x, y\ \ge \ a $ $$ \Gamma(x) + \Gamma(y) \le ...
8
votes
2answers
2k views

How to evaluate Riemann Zeta function

How do I evaluate this function for given $s$? $$\zeta(s) = \sum_{n=1}^\infty \frac1{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$$
9
votes
2answers
882 views

Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?

The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some properties of the gamma function: $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$ $\Gamma (1)=\Gamma ...
4
votes
2answers
599 views

Question Relating Gamma Function to Riemann Zeta function evaluated at integers

I was just reading a paper of Ramanujan entitled " On question 330 of Professor Sanjana" when i got stuck up with a Proposition which i am unable to answer. The proposition is if $ \displaystyle ...
89
votes
9answers
5k views

Why is Euler's Gamma function the “best” extension of the factorial function to the reals?

There are lots (an infinitude) of smooth functions that coincide with f(n)=n! on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^t dt$ is ...
8
votes
4answers
3k views

How to accurately calculate the error function erf(x) with a computer?

I am looking for an accurate algorithm to calculate the error function I have tried using [this formula] (http://stackoverflow.com/a/457805) (Handbook of Mathematical Functions, formula ...