# Tagged Questions

**0**

votes

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**2**answers

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 ...