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Strange Identities

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Post Thu Jan 09, 2014 12:49 pm
Shobhit Site Admin
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Posts: 850
Location: Jaipur, India

Hi all, I would like to share some curious/strange identities with you. Maybe in the future will be skilled enough to prove them?

This thread is not for proving identities but to simply state them. If you know any other crazy identities or want to comment don't hesitate to post.

1. \(\displaystyle \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)\cosh(10n+5)\pi}=\frac{1}{2} \sin^{-1} \left[\left\{ \left( 3-2\sqrt2\right)\left( 2+\sqrt5\right) \left( \sqrt{10}-3\right)\left( -\sqrt2+\sqrt[4]5\right)^2 \right\}^2 \right]\)

2. \(\displaystyle \sum_{n=0}^\infty (-1)^n \frac{(2n+1)^2}{\sinh (2n+1)\pi }=\frac{\sqrt{2}-1}{8}\frac{\pi^{3/2}}{\Gamma^6 \left(\frac{3}{4}\right)}\)

3. \(\displaystyle \prod_{k=1}^\infty \left(1+e^{-\frac{k\pi}{3}} \right)=\frac{e^{\frac{\pi}{72}}}{\sqrt[8]{2}}\sqrt[24]{\frac{\sqrt{2}+\sqrt[4]{3}}{(\sqrt{2}-\sqrt[4]{3})^5}}\)

4. \(\displaystyle \int_0^1 K(k)^3 \; dk = \frac{3}{1280\pi^2}\Gamma^8\left( \frac{1}{4}\right)\) where \(K(k)\) is the complete elliptic integral of the first kind.

5. \(\displaystyle \sum_{k=0}^\infty \frac{2k+1}{\left\{ 25+\frac{(2k+1)^4}{100}\right\}(1+e^{(2k+1)\pi}) }=\frac{4689}{11890}-\frac{\pi}{8}\text{coth}^2 \left(\frac{5\pi}{2} \right)\)

6.\(\displaystyle \int_0^\infty \frac{1}{\prod\limits_{n=0}^\infty (1+e^{-10 n \pi}x^2)}dx=\frac{\pi^{\frac{3}{4}}\Gamma \left(\frac{3}{4}\right)}{2e^{\frac{5\pi}{8}}}\sqrt{5}\sqrt[8]{2}\left( 1+\sqrt[4]{5}\right)\sqrt{\frac{1+\sqrt{5}}{2}}\)

Post Thu Jan 09, 2014 8:41 pm
galactus User avatar
Global Moderator
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Posts: 902
I posted this one under that Si thread. I thought it was worth noting.

Since \(\displaystyle \text{si}(x)=\int_{0}^{x}\frac{\sin(t)}{t}dt-\int_{0}^{\infty}\frac{\sin(t)}{t}dt=-\int_{x}^{\infty}\frac{\sin(t)}{t}dt\)

we have \(\displaystyle \text{Si}(x)-\frac{\pi}{2}=\text{si}(x)\)

The curious identity is

7. \(\displaystyle \text{si}(x)=-\int_{0}^{\frac{\pi}{2}}e^{-x\cos(t)}\cos(x\sin(t))dt\)

In other words: \(\displaystyle \text{Si}(x)=-\int_{0}^{\frac{\pi}{2}}e^{-x\cos(t)}\cos(x\sin(t))dt+\frac{\pi}{2}\)

Post Sat Jan 11, 2014 9:02 am
Shobhit Site Admin
Site Admin

Posts: 850
Location: Jaipur, India

Another integral which I have been trying to prove for a lot of time is this one:

8. \(\displaystyle \int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx=\frac{\pi}{8}\log \left( \frac{\pi^2}{8}\right)\)

But no matter how hard I tried I could not prove it.


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