Closed form of divergent infinite product? - MathOverflow

Closed form of divergent infinite product?

2010-06-21T20:06:42Z

Okay, we know that

$$ \frac{sin(x)}{x} = \prod_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\cdot\pi^2}\Big) $$ .

Is there some known (trigonometric(?)) function that is equal to the following infinite product?

$$ \prod_{n=1}^{\infty} \Big(1-\frac{x}{n\cdot\pi}\Big) $$

I'd be happy as well if someone could provide me with a function that is equal to a similar divergent infinite product (a function, for example, that is equal to 'my' inifite product, only $\pi=1$, or $x=x^2$, or something in that direction).

Thanks in advance,

Max Muller

Answer by coudy for Closed form of divergent infinite product?

2010-06-21T20:15:09Z

I would suggest the development of the Gamma function

$$1/\Gamma(z) = z e^{\gamma z}\ \Pi_{n=1}^\infty\ (1+{z\over n})\ e^{-{z\over n}}$$

Answer by Robin Chapman for Closed form of divergent infinite product?

2010-06-21T20:19:46Z

It's a divergent infinite product. You might as well ask for the sum of $$\sum_{n=1}^\infty\frac{x}{n\pi}.$$ You can "cure" the divergence by multipliying each term by a suitable factor, so for instance $$f(x)=\prod_{n=1}^\infty e^{x/n\pi}\left(1-\frac{x}{n\pi}\right)$$ does converge (as the $n$-th term is like $\exp(x^2/2n^2\pi^2)$). You can express this in terms of the gamma function which satisfies $$\frac1{\Gamma(x)}=x e^{\gamma x}\prod_{n=1}^\infty e^{-x/n}\left(1+\frac{x}{n}\right).$$ By using the identity $$f(x)f(-x)=\prod_{n=1}^\infty\left(1-\frac{x^2}{n^2\pi^2}\right)$$ one can deduce the identity $$\Gamma(x)\Gamma(1-x)=\frac\pi{\sin\pi x}.$$

Answer by castal for Closed form of divergent infinite product?

2010-06-21T21:21:22Z

Take a look at the first dozen pages of Andrews and Askey, which you can read online - http://books.google.com/books?id=nMm13WXpLt8C&lpg=PP1&dq=andrews%20askey&pg=PA1#v=onepage&q&f=false

Already on page 3, they give the product representation of 1/Gamma, which is essentially your function, modified to make it convergent.

On page 10, they treat the reflection formula, which shows that 1/Gamma is "half of the sine function", i.e it contributes the zeros on the negative x axis.