# 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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### Power series expansion of the lemniscate function

We consider the following function. $$u(x) = \int_{0}^{x} \frac{dt}{\sqrt{1 - t^4}}$$ $u(x)$ is defined on $[-1, 1]$. Since $u'(x) = \frac{1}{\sqrt{1 - x^4}} > 0$ on $(-1, 1)$, $u(x)$ is strctly ...
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### Compute $\lim_{x\to\infty} \frac{{(x!)}^{\frac{1}{x}-1} (x\Gamma(x+1) \psi^{(0)}(x+1)-x! \log(x!))}{x^2}$

What's the strategy one may use when facing a limit like this one? I think it's more important to know the possible ways to go than the answer itself. It's a problem that came to my mind again when I ...
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### Calculate $I(\alpha, x,y)=\int\limits_0^1 {{v^{\alpha - 1}}{{(1 - vx)}^{\alpha - 1}}{e^{vy}}dv,\,\,\,0 < \alpha ,x,y < 1}.$

I want to calculate this integral with singularity: $$I(\alpha, x,y)=\int\limits_0^1 {{v^{\alpha - 1}}{{(1 - vx)}^{\alpha - 1}}{e^{vy}}dv,\,\,\,0 < \alpha ,x,y < 1}.$$ I hope to obtain a ...
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### Different notions of q-numbers

It seems that most of the literature dealing with q-analogs defines q-numbers according to $$[n]_q\equiv \frac{q^n-1}{q-1}.$$ Even Mathematica uses this definition: with the built-in function QGamma ...
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$\alpha,\beta >0$ $$f(x,z)=\sum \limits_{n=0}^\infty \frac{e^{-\alpha n^2 x+\beta n z}}{n!}$$ $$\frac{\partial{f(x,z)}}{\partial z}=\beta \sum \limits_{n=1}^\infty \frac{e^{-\alpha n^2 x+\beta n z}... 0answers 257 views ### How to derive to inverse z transform of \sqrt{\frac{1-a^2}{1-\frac{a}{z}}} from Laguerre differential equation? How can I derive the inverse z-transform of:$$\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$$If Maple is not the way, how to derive manually? With Maple code I encounter some problems ... 1answer 113 views ### Minimal x for which \phi(k) > n for all k > x It's well-known that$$ \liminf_n\frac{\varphi(n)\log\log n}{n}=e^{-\gamma} $$and there exists an effective version$$ \varphi(n)>\frac {n}{e^\gamma\log\log n+\frac{3}{\log\log n}} $$valid for ... 0answers 190 views ### What is the correct differential equation for the Laguerre function? I would like to derive the correct Laguerre function from the differential equation but the differential equations seems different from the original one. What is the correct differential equation and ... 1answer 257 views ### Hypergeometric formulas for the Rogers-Ramanujan identities? Let q = e^{2\pi i \tau}. Given the j-function,$$j = j(q) = 1/q + 744 + 196884q + 21493760q^2 + \dots$$and define,$$k = j-1728$$Let \tau =\sqrt{-N}, where N > 1. Anybody knows how ... 0answers 88 views ### fastest way to evaluate \arg\zeta\left(\frac{1}{2}+i\text{t}\right)  [duplicate] Possible Duplicate: evaluation of  \operatorname{Arg}\zeta (1/2+is)  ?? If we consider$$\arg\zeta\left(\frac{1}{2} + i\text{t}\right) = \text{Im }\log\zeta\left(\frac{1}{2}+i\text{t}\right)...
Is the function $\hat{i}_0(x) = e^{-|x|} \sqrt{\frac{\pi}{2x}} I_{\frac{1}{2}}(x)$ positive or negative for negative $x$? $I_{\alpha}(x)$ above is a modified Bessel function. Here are my arguments. ...