# Tagged Questions

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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### Zeros of the lower incomplete gamma function

I'm interested in the zeros of the lower incomplete gamma function $$\gamma(s,x) = \int_0^x t^{s-1}e^{-t}\,dt \,.$$ In http://www.jstor.org/stable/2007135, Franklin (1919) derives some numerical ...
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### How do spectrums interact with bi-Lipschitz maps?

If it makes things simple, we can just stick to bi-Lipschitz maps from $S^k \rightarrow \mathbb{R}^d$ (w.r.t geodesic distance on the sphere with the standard round metric and the $2-$norm on the ...
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### An integral identity evaluating the gamma function

While reading a number theory paper I encountered the identity $$\int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$ ...
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### Calculate Ramanujan's class invariant by using modular equation of degree $5$

Let $$K(k):=\int_{0}^{\frac{\pi}{2}}\frac{d\phi}{\sqrt{1-k^2\sin^2\phi}}=\frac{\pi}{2}{ _2F_1\bigg(\frac{1}{2},\frac{1}{2},1;k^2 \bigg)}$$ where $0<k<1$ Let $K, K′, L$ and $L′$ denote the ...
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### Ask for a special function related to the error function

I am wondering whether anyone knows the following integration has a named special function or a reference $$F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y$$ for ...
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Is Rademacher complexity defined for any space of functions? Or are there restrictions on the function space over which this can be defined? For example is the Rademacher complexity defined or has ...
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### Hypergeometric function asymptotics

I came across the following hypergeometric function recently: $$_2F_1(1-n,p-2n+1;p-n+1;x)$$ where $p > 0$ is a non-integer constant, $n$ some large positive integer, and $x > 0$ a small ...
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### Division of half-integer Legendre functions of the second kind with different arguments

I'm in search of a formula for: $\frac{Q_{n-\frac{1}{2}}(\chi_1)}{Q_{n-\frac{1}{2}}(\chi_2)}= ??$ where I am hoping the result to be a function of $\frac{\chi_1}{\chi_2}$. Does anyone know of such ...
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### Combination of Generating Functions

Suppose I have the following generating functions: $$\frac{x^ke^{\left(z-\frac{1}{N}\right)x}}{N^{k-1}k!\sum_{j=0}^{N-1}w_N^{-jk}e^{\frac{w_N^jx}{N}}}=\sum_{j=0}^\infty H_{N,k,j}(z)\frac{x^j}{j!}$$ ...
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### What is the relationship between solutions for the parameterised second order differential equations

Let us consider the following parameterised complex-valued second order differential equations, and $u(x,\lambda)$ be the solution for $$u''+u'-i\lambda V(x)u=0, \, x\in [0,1],$$ What is the ...
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### Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$?

For naturals $n\ge m$, define $$I(n,m):=\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$$ with $\text{arcsinh}\ x=\ln(x+\sqrt{1+x^2} )$, so $\text{arcsinh} \frac12=\ln \frac{\sqrt{5}+1}2$. Is it ...
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I study some qualitative properties of Jacobian elliptic functions. Consider, for example, function $sn(u,k)$. In most applications, modulus $k\in(0,1)$ and then everything is very clear, since $sn(u,... 1answer 127 views ### An extreme of Jacobi elliptic function on an interval Consider the Jacobi elliptic function$sn(\cdot,k)$restricted to the interval$(0,2K)$, where$K=K(k)$is complete elliptic integral of the first kind. If$0<k<1$, then it is well known the ... 0answers 261 views ### Inverse Mellin of the exponential of the digamma function I'm looking for a function$f_p(x)$with real parameter$p>0$satisfying $$\int_0^\infty f_p(x)x^{s-1}dx=e^{-p\psi(s)}$$ where$\psi(s)$is the usual digamma function. The inverse Mellin formula ... 1answer 140 views ### Asymptotic behaviour of an integral For$k\in\mathbb{N}_{0}$and$x\in\mathbb{R}$, define $$I_{k}(x):=\int_{0}^{\pi/2}\cos(xg(\theta))\sin^{2k}\theta\,\mathrm{d}\theta$$ where$\$g(\theta)=\int_{\sin\theta}^{1}\frac{\mathrm{d}t}{\sqrt{(1-...
Motivation: I am trying to work on a problem related to computing the roots of a certain family of polynomials related to integer partition theory. In particular, I have been trying to Bridge the ...