# 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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### Is there a closed form for $\int_{-\infty}^{\infty} \frac{e^{-ax^2}}{\mathrm{erfc}{(-bx)}} dx$?

The integral expression is $$I = \int_{-\infty}^{\infty} \frac{e^{-ax^2}}{\mathrm{erfc}{(-bx)}} dx$$ where $a>0$ and $b>0$.
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### Evaluating $\int_{-\infty}^{\infty}x\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}^{2}\left(a\left(x-d\right)\right)\,\mathrm{d}x$

I have big difficulties solving the following integral: $$\int_{-\infty}^{\infty}x\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}^{2}\left(a\left(x-d\right)\right)\,\mathrm{d}x$$ I tried to ...
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### Does division of polynomials give an increasing function?

How can I show that f(a)=\frac{\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ i \\ \end{array} \right) \left(-1-\frac{1}{ar}\right)^i+1}{\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ ...
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### Ramanujan Summation

It seems that under the light of Ramanujan Summation the following is plausible: $$1 + {2^{2n - 1}} + {3^{2n - 1}} + \cdots = - \frac{{{B_{2n}}}}{{2n}}(\Re)$$ Alas, I can't really find any ...
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### What are special functions for?

If you read enough mathematics, you eventually come across several so-called "special functions". I'm always left wondering what on Earth these things are actually for. We have the Euler Gamma ...
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### A uniqueness proposition involving Erf, the error function

This is a MathOverflow cross-post (currently no answer there) and a generalization of a previous MathOverflow question, "Reducing system of equations involving Erf, Error Function". Consider the ...
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### Satisfying a Differential Equation and complex Laguerre

I have the following problem Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
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### What does $\langle Y_{lm} | Y _{\lambda\mu} \rangle = \delta_{l\lambda} \delta _{m\mu}$ mean?

In Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion, I can almost completely understand the recurrence relations described, but for one part. The $Y^l_m$ function is ...
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### Proving or disproving that if $\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$ both $a$ and $b$ are integers.

I found some formula about special function very complicated, so I am curious how you people solve this by hand. $$\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$$ but $a$ and $b$ are very ...
I just want to make clear of the definition of sinc(x). I know there is a normalized and unnormalized definition for the sinc function. If we have unnormalized sinc then we have: \sin(x)/x=\text{... 1answer 242 views ### Can it be shown that Y_0(\lambda_n a)J_0(\lambda_n a) - J_0(\lambda_n a)Y_0(\lambda_n a) \ne 0? Background I am currently looking into the task of describing a transient temperature field \theta(r,t) across the thickness a \leq r \leq b of an infinitely long and hollow cylinder exposed to a ... 1answer 407 views ### A Curious Binomial Coefficient Sum Let k, l \leq n be non-negative integers. Does the following identity simplify? \begin{align} \sum_{j = 0}^{k} \binom{k}{j} \binom{j + n -l + 1}{n} = \binom{n - l + 1}{n} \phantom1_{2}\mathsf{F}... 1answer 292 views ### What is the fractional derivative of the function \pi \cot (\pi x)? What is the fractional derivative of the function \pi \cot (\pi x)? I derived the following expression: (\pi \cot (\pi q))^{(p)}=-\frac{\zeta'(p+1,q)+(\psi(-p)+\gamma ) \zeta (p+1,q)}{\Gamma (-p)}... 1answer 269 views ### how to evaluate \int_0^1\sin(\frac{1}{x})dx? How can I evaluate the integral of\int_0^1\sin\left(\frac{1}{x}\right)dx.$$Maybe it needs the cosine integral to evaluate it, but I cannot understand it very well. Thanks a lot. 2answers 209 views ### Could you give a application of a special function on number theory or analysis? With the best effort i have ever taken, i couldn't find a application of a special function on number theory or analysis on the internet. By the way, why is the applications of special functions in ... 2answers 2k views ### Proof that \sum\limits_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x} regarding \zeta(3) and Apéry's proof I recently printed a paper that asks to prove the "amazing" claim that for all a_1,a_2,\dots$$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$and thus (probably) ... 1answer 437 views ### Was the definition of \mathrm{erf} changed at some point? I have seen two competing definitions of the error function. When I was an undergrad, Spiegel's Mathematical Handbook of formulas and tables (mine is the 1968 edition) was the definitive authority, ... 1answer 173 views ### Closed form of integral of \operatorname{erfc} \log t Is there any closed form expression for the following integral?$$ \int\limits_t^\infty \left(1- \operatorname{erf}(\log x) \right )dx $$or equivalently:$$ \int\limits_t^\infty \operatorname{erfc}...
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 ...