0
votes
1answer
111 views

Legendre differential equation with additional term

In an application I encountered the ODE $$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f \left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x \right) \right) \left( ...
9
votes
1answer
245 views

Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?

Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $x_n$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$. My ...
9
votes
1answer
813 views

Has anyone seen this series?

I come across the following infinite series. $$ \sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}. $$ In particular, I am interested in the case where $a=1/4$. ...
2
votes
1answer
165 views

Limit involving regularlized gamma function and its inverse

Let $$L(x)=Q\left(\frac{x}{2},\frac{a}{a+f(x)/\sqrt{x}}Q^{-1}\left(\frac{x}{2},1-b^{1/g(x)}\right)\right)$$ where $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}$ is the upper incomplete gamma function ...
4
votes
1answer
221 views

How to get an expression for this integral(Numerically/Analytically)

I have the following problem: I need to evaluate the integral $$\int_{\cos(\alpha)}^{1} P_l(t)P_{l'}(t) dt $$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P_l$ is the l-th ...
9
votes
2answers
482 views

The fraction of the sphere a fixed distance from a subspace

The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a ...
0
votes
0answers
243 views

Lambert W-function

I asked this question MSE, but didn't get any answers. Maybe here someone can help. I need to solve $$ \theta \rho^{\theta}+r \theta>v $$ where $\theta \in \mathbb{R}^{+}, -1 < r,v<1, \ ...
3
votes
0answers
458 views

Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions?

I am trying to somewhat simplify a series, whose coefficients feature generalised hypergeometric functions ${}_1F_2(1;a,a+1;z)$. I was unable to find useful functional relations for this specific ...
3
votes
2answers
579 views

Reducing system of equations involving Erf, Error Function

I have a system of equations: $$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$ $$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$ Where $x \le y$ and ${\rm Erf}$ is the Error Function. By ...