# Tagged Questions

Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

3answers
178 views

0answers
22 views

### Expressing spherical harmonics as a combination of other spherical harmonics

Spherical harmonics are a useful tool in physics, particularly in classic electrostatics and electrodynamics. Given an integer $l$, the spherical harmonic $Y_{l,m}$, where $-l\leq m\leq l$, solves the ...
1answer
489 views

1answer
63 views

0answers
5 views

### Invertibility of a polylogarithmic map

Consider a map defined on $\Bbb R\times(0,+\infty)$ and given by $$M:(a,b)\to(\rho,E),$$ $$\rho = \int_{\Bbb R^n}\frac{dx}{1+\exp(a+b|x|^2)}\\E=\int_{\Bbb R^n}\frac{|x|^2dx}{1+\exp(a+b|x|^2)}.$$ I ...
1answer
102 views

### Closed form of a “harmonic” alternating dilogarithm sum

Does the following sum $$S = \sum_{n\geq 2}(-1)^n \mathrm{Li}_2(2/n) = 1.14434\ 42096\ 91982\ 23727\ 39852\ 45805\ldots$$ have a closed form in terms of known constants? Neither the inverse ...
0answers
25 views

### How to simplify the following expression involving Jacobian elliptic functions?

I would like to show that a certain elliptic function $F(x)$ (that is periodic, say with some period $h$) has exactly two zeroes in $[0,h)$. Let us recall some notation. Given a parameter $m \in [0,1]$...
1answer
66 views

1answer
41 views

0answers
8 views

### proving the output of cantor pairing function is un correlated if one bit of input is changed

Pairing function $\pi :N \times N\rightarrow N$ is defined as: $\pi(a_{1},a_{2})=\frac{1}{2}(a_{1}+a_{2})(a_{1}+a_{2}+1)+a_{2}$. Exp 1. If $a_{1},a_{2}$ are input then output is $\pi_{1}$ Exp 2. If ...
4answers
51 views

### How to express $2x^3-x^2-3x+2$ as a linear combination of Legendre polynomials

I have used the formula \begin{align}p_0(x)&=1\\ p_1(x)&=x\\ p_2(x)&=\frac12(3x^2−1)\\ p_3(x)&=\frac12(5x^3−3x) \end{align} $$2x^3-x^2-3x+2=Ap_3(x)+Bp_2(x)+Cp_1(x)+Dp_0(x)$$ EDIT- \...