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The following mathematical equation was shown during the television show Fringe which aired on Friday, April 22nd. Any idea what it is?

Fringe formula

(edit by J.M.: for reference, this was Sam Weiss scribbling formulae in his notebook.)

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    I haven't watched: was that Walternate writing?2011-04-28
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    I appreciate this question. I think it's fun and lively - what is Fringe?2011-04-28
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    @mix: [This TV show](http://www.televisionwithoutpity.com/show/fringe/).2011-04-28
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    @J.M. : No, it was that bowling guy. He was observing a vortex and taking notes.2011-04-28
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    @Gunnar: Thanks! I won't be able to watch until tomorrow, that's why I asked. :)2011-04-28
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    *adds popular-math to followed tags*... +12011-04-28
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    Just a minor off topic of somewhat similar pop culture references: in the 21st episode of The Simpsons (season 22) there are the Kepler's laws of motion scribbled on a blackboard in two scenes. Pretty neat.2011-05-23
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    @BBischof, ditto.2012-03-30

3 Answers 3

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The last formula seems to be an integral expression for the Dirichlet $\eta$ function:

$$\eta(s)=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^s}\quad \Re(s) > 0$$

Using the relationship with the usual Riemann $\zeta$ function

$$\eta(s)=(1-2^{1-s})\zeta(s)$$

and this integral expression, you get that integral in the notebook:

$$\eta(s) = \frac1{\Gamma(s)} \int_0^{\infty}\frac{x^{s-1}}{e^x+1}\mathrm dx$$

There is also this closely related integral (which Arturo mentions in his answer).

The (first part of the) second line looks to be the chain of relations relating Riemann $\zeta$, Dirichlet $\eta$, and Dirichlet $\lambda$:

$$\frac{\zeta(s)}{2^s}=\frac{\lambda(s)}{2^s-1}=\frac{\eta(s)}{2^s-2}$$

In the second part, the expressions look to be the differentiation of Dirichlet $\eta$, but the screenshot is fuzzy around that region...

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    Ok. Now that I look closely, it is a $e^x +1$, not a $e^x-1$. +1.2011-04-28
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    A confession: I was actually looking at the left hand side, and it sure looked like an $\eta$ to me... as for the equations on the second line, check out formula 3 in the MathWorld link I gave.2011-04-28
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    It is quite hard to read the second line, so I actually ignored those. Now that you point out, it does look like Eqn (3) :-)2011-04-28
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    Given the R(s) > 1/2 above that last integral, I wonder if the show was trying to show him solving (or attempting to solve) the Riemann Hypothesis...2011-05-23
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    The succeeding scene had Sam writing a 0 after the =, so I suppose yes.2011-06-02
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By sheer coincidence, this equation occurs in P. Mark Kayll's paper Integrals don't have anything to do with discrete Math, do they? (Mathematics Magazine 84 no. 2, April 2011, pages 108-119, doi:10.4169/math.mag.84.2.108, which I was reading through today. It is equation (6) on page 111.

$\zeta(s)$ is the Riemann zeta function, $$\zeta(s) = \sum_{k=1}^{\infty}\frac{1}{k^s}.$$

$\Gamma(s)$ is the Gamma function, $$\Gamma(s) = \int_0^{\infty} t^{s-1}e^{-t}\,dt.$$

$R(s)\gt \frac{1}{2}$ says that the real part of $s$ is greater than $\frac{1}{2}$.

The final equation is just a recasting of the equality $$\zeta(x)\Gamma(x) = \int_0^{\infty}\frac{t^{x-1}}{e^t-1}\,dt$$ which holds whenever $\Gamma(x)$ is finite.

Nothing to get too excited about... unless you don't know what any of the symbols mean, which I suspect holds for a very large portion of the viewership of Fringe.

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    Actually it is $e^x + 1$, not $e^x -1$ (sure looked like that to me too!), so J.M's answer is more accurate. (And the symbol and $Re(s) \gt 1/2$ fit better for that I suppose).2011-04-28
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    Arturo, [the DOI doesn't seem to work](http://dx.doi.org/10.4169/math.mag.84.2.108). Maybe link to [JSTOR](http://www.jstor.org/pss/10.4169/math.mag.84.2.108) instead?2011-04-28
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    @J.M., also available at http://www.umt.edu/math/reports/kayll/Karply-MMag-submit09.pdf2011-04-28
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    @lhf: Thanks! I shall read it now...2011-04-28
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    @Moron: Hmmm... I can see how it might seem like a $+$ rather than a $-$; but the vertical stroke seems much lighter than the rest.2011-04-28
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    @J.M. I suspect the doi has not been created yet; the issue *just* came out in the past couple of weeks.2011-04-28
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    @Arturo: I guess I was under the impression that the DOI guys are prompt... guess not. :)2011-04-28
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    @J.M. I did double check; this is the DOI listed on the bottom of the first page of the article.2011-04-28
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It's Riemann's zeta-function as a Mellin transform. See http://en.wikipedia.org/wiki/Riemann_zeta_function#Mellin_transform

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    Oh, it's $+1$, not $-1$. And it's $\eta$, not $\zeta$...2011-04-28