Articles of special functions

Fourier transform of 3D Sinc function

What is the Fourier trasnform of the function $$\frac{\sin(P|\mathbf{x-y}|)}{|\mathbf{x-y}|}$$ where $P$ is a real parameter and $\mathbf{y}$ is a fixed point in three-dimensional space?

partial sum of Basel problem related to series involving Beta function

I ran across a series and got to wondering how this is so. We are all familiar with the famous $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^{2}}=\frac{{\pi}^{2}}{6}$ But, how can we show: $\displaystyle\sum_{k=1}^{n}\frac{1}{k^{2}}=\frac{{\pi}^{2}}{6}-\sum_{k=1}^{n}\frac{\beta(k,n+1)}{k}$ where $\beta$ is the beta function. Apparently, $\displaystyle\sum_{k=1}^{n}\frac{\beta(k,n+1)}{k}={\psi}^{'}(n+1)$ somehow. But, the above beta series can be written $\displaystyle\sum_{k=1}^{n}\frac{1}{k}\int_{0}^{1}x^{k-1}(1-x)^{n}dx$. Also, ${\psi}^{'}(n+1)=\displaystyle\sum_{k=0}^{\infty}\frac{1}{(n+k+1)^{2}}$ I know that ${\psi}(x)=\int_{0}^{1}\frac{t^{n-1}-1}{t-1}dt-\gamma$ Maybe differentiate w.r.t n […]

Elliptic integrals with parameter outside $0<m<1$

I’m attempting to implement an equation (for calculating magnetic forces between coils, eqs (22–24) in the linked paper) that requires the use of elliptic integrals. Unfortunately these equations require the evaluation of the elliptic integrals far outside their standard parameter range of $0\le m\le 1$ and the numerical implementations I have available to evaluate them give […]

Integral of product of exponential function and two complementary error functions (erfc)

I found the following integral evaluation very interesting to me: Integral of product of two error functions (erf) and I hoped that I could use that result to evaluate the following integral: $$ \int_{-\infty}^{\infty}\exp\left(-t^{2}\right)\,\mathrm{erfc}\left(t-c\right)\,\mathrm{erfc}\left(d-t\right)\,\mathrm{d}t=\frac{4}{\pi}\int_{-\infty}^{\infty}\exp\left(-t^{2}\right)\int_{t-c}^{\infty}\int_{d-t}^{\infty}\exp\left(-u^{2}-v^{2}\right)\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}t $$ So I note that $u\geq t-c$, $v\geq d-t$, thus $t\leq u+c$ and $t\geq d-v$, thus $d-v\leq t\leq u+c$ and $u+v\geq […]

Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$

I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer. I am interested to find the bounds on the value it can take or an approximation for it. Since $$0<\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}<1, $$ I was thinking that ${{e}^{z}}$ would be an upper bound. Is […]

Special functions related to $\sum\limits _{n=1}^{\infty } \frac{x^n \log (n!)}{n!}$

While doing some caculation related to von Neumann entropy, I encountered this kind of convergent series. $$\text{Exl}(x) \equiv \sum _{n=1}^{\infty } \frac{x^n \log (n!)}{n!}$$ In my calculation, this function Exl$(x)$ appears in some places where exponential function should be, for example, $$\frac{\cosh (x) \text{Cxl}(x) + \sinh (x)\text{Sxl}(x)}{\cosh(2x)}$$ appears in my calculation, where $$\text{Cxl}(x) \equiv \frac{\text{Exl} […]

Bernoulli number type asymptotics

I find an interesting formula but I can not prove it. Show that $$I_n=(-1)^{n+1}\int_0^1 B_{2n+1}(x)\cot(\pi x) \, dx\sim\frac{2(2n+1)!}{(2\pi)^{2n+1}}$$ where $B_n(x)$ is the Bernoulli Polynomials.

Real roots plot of the modified bessel function

Could anyone point me a program so i can calculate the roots of $$ K_{ia}(2 \pi)=0 $$ here $ K_{ia}(x) $ is the modified Bessel function of second kind with (pure complex)index ‘k’ 😀 My conjecture of exponential potential means that the solutions are $ s=2a $ with $$ \zeta (1/2+is)=0. $$

Why the sum of the squares of the roots of the $n$th Hermite polynomial is equal to $n(n-1)/2$?

How to prove that the sum of the squares of the roots of the $n$th Hermite polynomial is $\frac{n(n-1)}{2}$? I tried with Vieta formulas, but it’s hard. I appreciate a proof or reference to it. An idea is to use the definition of sum of Hermite polynomials, but do not know.

Estimating the Gamma function to high precision efficiently?

I know there are several approximations of the Gamma function that provide decent approximations of this function. I was wondering, how can I efficiently estimate specific values of the Gamma function, like $\Gamma (\frac{1}{3})$ or $\Gamma (\frac{1}{4})$, to a high degree of accuracy (unlike Stirling’s approximation and other low-accuracy methods)?