How to prove Gauss’s Multiplication Formula?
$$\Gamma(nz)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left(z+\frac{k}{n}\right)$$
Any help like an answer or link would be appreciated. Thanks for all help.
Asked
Active
Viewed 4,609 times
1
-
2See Clasical Analysis by E. Chiang. The statement of Gauss's Multiplication Theorem is on page 45 and the proof begins on the next. – Gamma Function Aug 04 '13 at 20:21
-
@JacobMayle thanks alot – mnsh Aug 04 '13 at 20:27
-
@JacobMayle Me too, a big help. – Feb 12 '14 at 21:37
1 Answers
0
I think the proof along Ahlfors' lines would be to differentiate both side and use the identity $$ \frac{d}{dz}(\frac{\Gamma'(z)}{\Gamma(z)})=\sum^{\infty}_{n=0}\frac{1}{(z+n)^{2}} $$ on both sides of the equation. It is easy to verify that $\Gamma(z)$ and $\prod^{n-1}_{i=0}\Gamma(\frac{z}{n}+\frac{i}{n})$ has the same poles, and taking log derivative on both sides give you an extra $n^{2}$ factor. So all we need is to fix the constant and the $e^{az+b}$ term. But at this point I do not know how to derive it rigorously, and I think the textbook others suggested might help.
I found there is a direct proof here:
Bombyx mori
- 19,638
- 6
- 52
- 112