<em id="zlul0"></em><dl id="zlul0"><menu id="zlul0"></menu></dl>

<em id="zlul0"></em>

<dl id="zlul0"></dl>
<div id="zlul0"><tr id="zlul0"><object id="zlul0"></object></tr></div>
<em id="zlul0"></em>

<div id="zlul0"><ol id="zlul0"></ol></div>

# 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.

561 questions
Filter by
Sorted by
Tagged with
0answers
47 views

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{...
2answers
343 views

### 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 $z$. ...
0answers
140 views

### 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 $\operatorname{li} (x)$ ...
1answer
71 views

1answer
36 views

### 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 $1\leq k\leq n\leq m$, ...
1answer
161 views

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_{...

15 30 50 per page
山西福彩快乐十分钟
<em id="zlul0"></em><dl id="zlul0"><menu id="zlul0"></menu></dl>

<em id="zlul0"></em>

<dl id="zlul0"></dl>
<div id="zlul0"><tr id="zlul0"><object id="zlul0"></object></tr></div>
<em id="zlul0"></em>

<div id="zlul0"><ol id="zlul0"></ol></div>
<em id="zlul0"></em><dl id="zlul0"><menu id="zlul0"></menu></dl>

<em id="zlul0"></em>

<dl id="zlul0"></dl>
<div id="zlul0"><tr id="zlul0"><object id="zlul0"></object></tr></div>
<em id="zlul0"></em>

<div id="zlul0"><ol id="zlul0"></ol></div>
新时时走势图 安徽时时计划软件 体育票广东时时 广东时时交流群 全天极速时时计划软件 北京时时计划预测 新彊时时三星走势图 体彩走势图下载 幸运飞艇走势图手机版 时时彩五星一码计算法