# 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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### Closed-form solution to $\frac{\ln x}{x} = k$

What is the solution in $x$ to $$\frac{\ln x}{x} = k ?$$ I suspect this has something to do with the Lambert W function, since that's used in solutions of the form $x\ln(x) = k$, but the Wikipedia ...
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### Does asymptotic expansion of Whittaker function $W_{\lambda , \mu}(z)$ exist for $|\lambda| \to 0$?

Suppose Whittaker function $$\tag 1 W_{\lambda , \mu}(z)$$ Does some asymptotic expansion exist for the case $|\lambda| \to 0$? I'm interested not in the case of $\lambda = 0$, but in the case of ...
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### Almost periodic function vs quasi periodic function

I am doing some work regarding quasi periodic function but I am not able to figure out the difference between almost periodic and quasi periodic functions.Can anyone let me know about it? Thanks ...
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### The expression of the sum of infinite gaussian functions

Let $f(x|\mu,\sigma^2)$ be the gaussian function (normal distribution): $$f(x|\mu,\sigma^2)=\frac{1}{\sigma\sqrt{2\pi}}e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$$ We know its integral over $\mathbb{R}$ is ...
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### general procedure for contour integration of $\int_{0}^{\infty} \mathrm{Ai}(x)^{n} dx$

In Richard Crandall's On The Quantum Zeta Function, following eq. 4.11: $$\int_{0}^{\infty}\mathrm{Ai}(x)^{2}dx=\frac{\Gamma(\tfrac{5}{6})}{2\pi^{5/6}12^{1/6}}$$ “again derivable by contour ...
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### The domain of the Digamma function and its extension

First, we know that $\Gamma(x)>0$, for all $x>0$, so define $\psi(x)=\frac{\mathrm{d} }{\mathrm{d} x}\ln(\Gamma(x))=\Gamma'(x)/\Gamma(x)$, for all $x>0$. This is the Digamma function. It is ...
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### Definite integral $\int_{0}^{\infty} e^{-a t} \log(t)\log(1+t)\,dt$

Is there a closed-form expression (possibly in terms of special functions) for the integral: $$\int_{0}^{\infty} e^{-a t}\log(t)\log(1+t)\,dt,$$ where $a>0$?
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### About the domain of the Gamma function

I started to read about the history of the Gamma Function. There are three places I liked most, The early history of the factorial function (p. 239 - 243) Leonhard Euler's Integral: An Historical ...
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### Name of a particular improper integral

I am curious if there is a particular name for this, $\int\limits_{-\infty}^\infty e^{i\xi^2}d\xi$. I think it might be related the Fresnel integral but I cannot see it, any suggestions?
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### Asymptotic limit of the following integral?

I am interested in the asymptotic limit of the following integral for $a\rightarrow\infty$, $$\int_0^1\mathop{\mathrm{d}x}J_2(ax)x^n,$$ where $n>-1$ and $J_2(x)$ is the Bessel function of first ...
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### Legendre Polynomial definite integral identity

I'm doing a problem involving legendre polynomials and I got stuck in this integral: $$I_k=\int_{-1}^{1} x P_{2k+1}(x)dx$$ Update: Note that the function in the integral is even If $k=0$, then ...
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### Differential equation with variable coefficients

I saw this differential equation somewhere $$y''+xy=0$$ it was solved using the substitution $$y=x^\alpha u$$ where $\alpha$, is a constant. My question is how can we substitute for $y$ with an ...
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### Does anyone know a function that can describe a harmonic series?

I want to find a function that satisfies the following functional equation: $F(z+1)=1/z+F(z)$ This is a generalization of harmonic series 1 + 1/2 + 1/3 + 1/4 + ...,...
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### On Hyper-geometric Functions and its recurrence relation

I research in generating functions of Hyper-geometric functions $_2F_1(a+n,b;c+n;x)$ using Lie group theoretic method and so the recurrence relation is important in this method. I want recurrence ...
### Want to check that $\sum_{j=0}^{k-1}w^{ jm}=0$, $m\not\equiv 0 \pmod{k}$ where $w=e^{2\pi i/k}$
If $f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$, then $$\sum_{n=0}^{\infty}a_{kn+m}x^{kn+m}=\frac{1}{k}\sum_{j=0}^{k-1}w^{-jm}f(w^j x) \tag{1},$$ where $w=e^{2\pi i/k}$ is a primitive $k$th root of ...