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# Questions tagged [special-functions]

Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.

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Let $\mu$ be the Gaussian measure $d\mu(x) = e^{-x^2/2} \frac{dx}{\sqrt{2\pi} }$. I am interested in the following random matrix integral defined for all $s \in \mathbb{R}$, $N \geq 1$ and $a ... 0answers 112 views ### A complex integration formula I'm trying to integrate a formula but I can’t figure it out. Its calculation involves an error function. Here's the formula:$f(a, b, c)=\int_{0}^{+\pi} d \theta \exp (a \cos \theta) \operatorname{...
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### Сlosed formula for $(g\partial)^n$

The objective is to obtain a closed formula for: $$\boxed{A(n)=\big(g(z)\,\partial_z\big)^n,\qquad n=1,2,\dots}$$ where $g(z)$ is smooth in $z$ and $\partial_z$ is a derivative with respect to $z$. ...
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### Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral

It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$ and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$ where $\operatorname{li} (x)$ ...
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### Joint density of a quadratic function of entries of orthogonal matrix

$U=(U_{ij})_{1\leq i,j\leq m},V=(V_{ij})_{1\leq i,j\leq m}$ are independently and uniformly distributed on the orthogonal group $O(m)$. For any positive integer $k,n$ such that $1\leq k\leq n\leq m$, ...
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In (1) there is a property of spherical Bessel functions, which's derivation I can not find in the literature. ${\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}^{2}}\left(z\right)=\sum_{k=% 0}^{n}\... 0answers 35 views ### Specify modified error function in form of error functions How can we express$\mathrm{erf}(\frac{t-a}{m})$as a sum of functions of the form$\mathrm{erf}(t)\$? I am developing a fitting routine and I encounter this integral: $$\int_{0}^{x}\mathrm{erf}\left(\... 0answers 55 views ### Differential equation with Fresnel integral We have \frac{y'(x)}{\cos(x)}=C(x) and need to find y(x). Generally we should express y(x) through C(x) and elementary functions. I can only do it through C(x) and S(x), or through \Phi(x).... 0answers 51 views ### Positivity and zeros of Heun's function I am interested in understanding where in the complex plane a Heun function might vanish, or where it (or its real part) is positive. Consider the case where the Frobenius indices at 0 are (0,1- \... 2answers 92 views ### Non-asymptotic upper bound of right tail of Gamma function I'm wondering if there is any non-asymptotic upper bound for the following Gamma function:$$f_a(x)=\int_{x}^{\infty}t^a\exp(-t)dt$$for x>0,a>0? Something like x^a\exp(-x)? 1answer 91 views ### Ratio of hypergeometric function Given a>b>0, is there any upper bound of the following ratio of hypergeometric function?$$\frac{_2F_1(a,1-b;a+1;x)}{_2F_1(a,1-b;a+1;y)}$$for 1>x>y>0 ideally in the form like some ... 1answer 168 views ### Ratio of Selberg integral I'm considering a ratio of incomplete Selberg integral:$$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...

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