What is the first formula in "Proof" and how can I prove it?
Euler's Reflection Formula - ProofWiki
Thank you...
What is the first formula in "Proof" and how can I prove it?
Euler's Reflection Formula - ProofWiki
Thank you...
If as 'first formula' You mean that...
$\displaystyle \sin \pi z = \pi\ z\ \prod_{n \ne 0} (1-\frac{z}{n})\ e^{\frac{z}{n}$ (1)
... then that's a perfect rubbish and I suggest You don't waste time trying to 'demonstrate' it...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
I came up with idea, tell me if that is wrong....
We need to show that:
$\displaystyle \sin \pi z = \pi\ z\ \prod_{n \ne 0} (1-\frac{z}{n})\ e^{\frac{z}{n}$
If we know that $\displaystyle \sin \pi z = \pi\ z\ \prod_{n \ne 0} (1-\frac{z^2}{n^2})$, then we can re-write it in the form:
$\displaystyle \pi\ z\ \prod_{n \ne 0} (1-\frac{z^2}{n^2})=\pi\ z\ \prod_{n \ne 0} (1-\frac{z}{n})(1+\frac{z}{n})$
And the question now is:
Is it true that,
$\displaystyle \prod_{n \ne 0} (1+\frac{z}{n})= \prod_{n \ne 0} e^{\frac{z}{n}}$
I said the the formula was 'a perfect rubbish' because it is written in ambigous form... the usual and fully clear formula is...
$\displaystyle \sin \pi\ z = \pi\ z\ \prod_{n=1}^{\infty} (1-\frac{z^{2}}{n^{2}})$ (1)
The 'demonstration' of (1) is done using the Weierstrass factorisation theorem...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$