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Results tagged with closed-form-expressions
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user 113397
For questions that specifically ask for determining a closed form of equations, integrals etc.
1
vote
1
answer
248
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On finding the critical points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+...
Given some constant $a\in\mathbb{R}-\{0\}$, find $x_0$ such that $f'(x_0)=0$ where $$f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x.$$
I have managed to write $f'(x_0)=0$ as $$\ …
- 1,459
5
votes
Accepted
Analytic expression for $\int_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\p...
We seek \begin{align}I&=\int_0^\infty\frac{e^{-st}\sin t}{t(e^{at}+1)}\,dt\\&=\sum_{n\ge0}(-1)^n\int_0^\infty\frac{e^{-(s+a(n+1))t}\sin t}t\,dt=-\sum_{n\ge1}(-1)^n\arctan\frac1{s+an}\end{align} which …
- 1,459
2
votes
New series for $\pi$ from string theory
It is unlikely $S(1)$ has a closed form.
We have \begin{align}S(1)&=4\sum_{k\ge1}(-1)^k\binom{-1/(4k)}k\frac{2k}{2k+1}\end{align} and (substituting $\lambda=0$ in the original Saha-Sinha formula for $ …
- 1,459