Anyone Watch Fringe - Mysterious Equation

[erok81]

So I was watching Fringe this morning and this came up. Then the character ominously solved it with a zero. I could solve the integral but there is a symbol that appears in the beginning that I am not familiar with. Plus, with the emphasis on him solving it with zero, maybe this is a famous equation and something bad happens when it equals zero.

Here is the formula in question (the best I can see it anyway).

$$n(s)=\frac{1}{\Gamma (s)} \int_{0}^{\infty} \frac{x^{s-1}}{e^{x}+1}dx$$

And here is a screen shot of his notepad. Which includes a few other things.

 

[jhae2.718]

$$\Gamma (s)$$ is the Gamma function, which is an extension of the factorial function to the reals and complex numbers.

 

[pergradus]

Wtf is the gamma function doing on fox?

 

[Leptos]

Wtf is the gamma function doing on fox?

Satisfying a recursion formula I suppose

 

[erok81]

Sounds like they just picked something that looked cool. I was hoping for an actual something, but I guess not.

 

[ideasrule]

You can easily show that the equation cannot be 0 for any value of s, because the integrand is always non-zero. The only exception is for the poles of the gamma function (s=0, -1, -2...).

By the way, if e^x+1 were instead e^x-1, n(s) would be a definition of the Riemann zeta function. The previous line is R(s)>1/2, which makes me think this has something to do with the Riemann hypothesis.

 

[jhae2.718]

It's probably supposed to be the zeta function.