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 November 30th, 2012, 06:18 AM #1 Math Team     Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions Some special functions properties and constants. My favourite List : $\Gamma (z)= \int^\infty_0 \, e^{-t}\cdot t^{z-1}\, dt$ $\Gamma (z+1)\,=\,z\Gamma(z)$ $\int^1_0 (1-t)^n \cdot t^{z-1}\,dt\,=\, \frac{n!}{z_{n+1}}\,\,\,\,\,\,z_n=z(z+1)\cdot\cdot \cdot (z+n-1)$ $\beta(x,y)\,=\,\int^\infty_0 \frac{t^{x-1}}{(1+t)^{x+y}}\, dt\,=\, \int^1_0 t^{x-1}\cdot(1-t)^{y-1}\,dt\,=\, \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\,=\, 2\int^{\frac{\pi}{2}}_0 \, \cos ^{2x-1}(t) \cdot \sin^{2y-1}(t) \,dt$ $\Gamma(\frac{1}{2})\,=\, \int^{\infty}_{-\infty}\, e^{-x^2}\,dx\,=\, \sqrt{\pi}$ $\Gamma(z)\Gamma(1-z)\,=\,\frac{\pi}{\sin(\pi z)}$ $\Gamma(z) \Gamma(z+\frac{1}{2})\,=\, 2^{1-2z}\,\sqrt{\pi}\, \Gamma(2z)$ $erf(z)\,=\, \frac{1}{2 \pi }\int^{z}_{0}\,e^{-t^2}\,dt\,$ $Ei(z)\,=\,\int^{\infty}_{z}\,\frac{e^{-t}}{t}\,dt\,=\,\gamma\,+\, \log|z|\,+\, \sum^{\infty}_{0}\frac{z^n}{n\cdot n!}$ $\mathcal {L}(f(t))(s)\,=\, \int^{\infty}_0\,e^{-st}\,f(t)\, dt$ $\zeta(z)\Gamma(z)\,=\,\int^{\infty}_0\frac{t^{z-1}}{e^t\,-\,1}\,dt$ $\zeta(z)\,= \sum_{n=1}^\infty\frac{1}{n^z} = \prod_{p \text{ prime}} \frac{1}{1-p^{-z}}$ $G\,=\,\sum^{\infty}_{0}\frac{(-1)^n}{(2n+1)^2}\,=\,\int^1_0 \frac{\arctan(t)}{t}\,dt$ $\gamma \,=\, \lim_{n \rightarrow \infty}\sum_{k=1}^{n}$\frac{1}{k}\,-\,\log(n)$$ [color=#400040]If you spot any mistake please inform me. Feel free to post any new function, identity or property.[/color]
 December 1st, 2012, 12:29 AM #2 Senior Member   Joined: Aug 2011 Posts: 334 Thanks: 8 Re: Some special functions properties and constants. My favourite list is in the paper "Safari on the contry of the special functions" http://www.scribd.com/JJacquelin/documents
 December 29th, 2012, 12:25 AM #3 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Some special functions properties and constants. The Hypergeometric function and the Weierstrass Theta function are, maybe, enough interesting to make a good position in your list .
December 29th, 2012, 12:44 AM   #4
Math Team

Joined: Aug 2012
From: Sana'a , Yemen

Posts: 1,177
Thanks: 44

Math Focus: Theory of analytic functions
Re: Some special functions properties and constants.

Quote:
 Originally Posted by mathbalarka The Hypergeometric function and the Weierstrass Theta function are, maybe, enough interesting to make a good position in your list .
I will consider looking at them ,, Thanks for the advice .

 January 31st, 2013, 06:01 AM #5 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Some special functions properties and constants. I will suggest adding three more important special constants in mathematics : Brun's constant, Meissel-Mertens constant and the famous Twin prime constant. All the three constant has an analytic representation.

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