# Tagged Questions

Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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### An inequality from the handbook of mathematical functions (by Abramowitz and Stegun)

Prove that $$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$
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### Identity concerning $e^{ia\sin{x}}$ as a series of bessel functions

Prove the following identity: $$e^{ia\sin{x}}=\sum_{-\infty}^{+\infty}J_k(a) e^{ikx},$$ where $a$ is a real constant and $J_k$ is the Bessel function of the first type of ...
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### Does this series converge (squares of associated Legendre polynomials)?

Consider the following series (where $l,\,m\in\mathrm{Z}\,$): $S = \displaystyle\sum^{\infty}_{l\,=\,2} \frac{2l+1}{(l-1)(l+2)(1+l^2)}\sum^{l}_{m\,=\,-l}\frac{(l-m)!}{(l+m)!}\Big(P^m_l(x)\,\Big)^2$, ...
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### Bounds on geometric sum

Consider the sum $\sum_{x=1}^{\infty} \frac{\log{x}}{z^x}$. We can assume that $z\geq1$ (and is real). Mathematica gives this sum as -PolyLog^(1, 0)[0,1/z] ...
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### for what $\nu$ does Riemann-Liouville differintegral of digamma function $\psi(z)$ exist?

For what values of $\nu$ does the Riemann-Liouville differintegral $_{-\infty}D_{z}^\nu$ of the digamma function $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$ exist, with $c=-\infty$? All I've got so far is ...
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### Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p}$ been discussed anywhere?
Can anyone prove this (I'm very confident that it is correct) or have any idea how this can be handled:  \lim_{n \rightarrow \infty} \frac{1}{n-1}\sum_{i=1}^{n-1} \frac{1}{(\alpha-1)(n-i) -1} \frac{...