# 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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### relationship between solution of quintic in terms of $_{4}F_{3}$ hypergeometric function and theta functions

There is one approach (Bring radical/method of differential resolvents) to the general solution to the quintic that gives the solution for a particular root $v\in\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ in ...
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### Special case of Meijer G function

I have an instance of the Meijer G function (using the definition from http://en.wikipedia.org/wiki/Meijer_G-Function, first equation there) that seems like, given its simplicity, it should be ...
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### complete elliptic integral of the first kind

I'm looking for any "closed form" for the coefficient of the $\ell$-th power of $K(x)$, the complete elliptic integral of the first kind. Thanks.
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### Power series $x f''(x) + f'(x) + xf(x) = 0$

Find a power series with radius of convergence $R = \infty$ such that $$f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$$ satisfies $$x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R.$$ How ...
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### Deriving the form of the Exponential Integral from a given integral

The Wikipedia entry on Asymptotic Expansion outlines a detailed example, where it refers to the fact that the integral $$\int_0^\infty \frac{e^{-w/t}}{1-w} \, dw$$ ...
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### Are there asymptotic expressions for multiple zetas $\small \zeta(s),\zeta(s,s),\zeta(s,s,s),\ldots$ where $\small s=1+\delta, \delta\to 0$?

Playing around with elementary symmetric functions I tried to generalize that to infinite series and arrived at the well known concept of MZV ("multiple zeta values"). At the moment I'm only ...
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### Are there well known lower bounds for the upper incomplete gamma function?

Let $a >0, b >0$, and $r \in \mathbb{R}$. I am trying to find a lower bound for the integral $$\int_a^\infty y^{-r} \exp\left( - b(y-a)^2\right) \,\mathrm dy.$$ After consulting the Wikipedia ...
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### Connection between Legendre polynomial and Bessel function

In Abramovitz and Stegun (Eq. 9.1.71) I found this curious relation $$\lim_{\nu\to\infty} \left[ \nu^\mu P_\nu^{-\mu}\left(\cos \frac{x}{\nu} \right) \right]= J_\mu(x) \qquad(1)$$ valid for $x>0$. ...
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### Fourier Transform of Bessel function with square root argument

Fourier Transform of the following function: ...
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### Evaluating $\zeta(0)$ using the functional equation of Riemann-Zeta function.

$$\zeta(it)=2it\pi it−1\sin(i\pi t/2)\Gamma(1−it)\zeta(1−it).$$ Everything on the RHS is never zero, Does that means LHS has no zeros, since $\sin(s)$ has a simple zero at $s=0$ while $\zeta(1−s)$ ...
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### Inequality involving the regularized gamma function

Prove that $$Q(x,\ln 2) := \frac{\int_{\ln 2}^{\infty} t^{x-1} e^{-t} dt}{\int_{0}^{\infty} t^{x-1} e^{-t} dt} \geqslant 1 - 2^{-x}$$ for all $x\geqslant 1$. ($Q$ is the regularized gamma function.) ...
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### Extending partial sums of the Taylor series of $e^x$ to a smooth function on $\mathbb{R}^2$?

Is there a smooth function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,n)$, where $n\in\mathbb{N}$, is the truncated Taylor series of $e^x$, namely $1+ x + \frac{x^2}{2} + \dotsb + \frac{x^n}{n!}$, ...
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### Integral of digamma function

I was attempting to evaluate a series $$\sum_{n=1}^\infty \frac{1}{n} \ln\left(1+\frac{1}{n}\right)$$ Since $$\frac{1}{n}\ln\left(1+\frac{1}{n}\right)=\int_0^1 \frac{1}{n(n+t)}dt,$$ I rewrote it as ...