# Tagged Questions

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

14 views

### Derivative of Incomplete Gamma Function

For the following incomplete Gamma function: $$Γ(1+d,A-c \ln x)=\int_{A-c\ln x}^{\infty}t^{(1+d)-1}e^{-t}dt$$ I am trying to calculate the derivative of $Γ$ with respect ...
11 views

### Solving the definite integral $\int_{-\psi/2}^{\psi/2}\,\exp{(A\sin(\theta+\phi))}d\theta$

I need to solve this definite integral: $$\int_{-\psi/2}^{\psi/2}\,\exp{(A\sin(\theta+\phi))}d\theta$$ where $A$ is a real positive constant and $\psi\in[0,2\pi]$. I know that for $\psi=2\pi$ the ...
97 views

### finding the series $\sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$

My goal is to solve this series $$S(x) = \sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$$ I did took the derivative first w.r.t $x$ $$S'(x) = \sum_{n=1}^\infty \frac{x^{n-1}}{n!}$$ which I ...
44 views

### Solve the nth zero of a function. [on hold]

Say I have a mystery continuous function, could be anything. f(x) Assuming we don't know the distribution of the zeros of the function, Is there a known way to solve the nth zero (hits the x-axis)? ...
28 views

### Series expansion of elliptic integral involving n th order polynomial in the denominator

My goal is to find an expansion in powers of 1/ρ of the integral: $$I_n(\rho)=\int_\rho^{+\infty}\frac{dt}{(E_n(t))^2\sqrt{t^2-h_2^2}\sqrt{t^2-h_3^2}},\quad \rho \ge h_2$$ ...
25 views

### Another Laplace transform of a function with square roots.

This question is very much related to this (one). Let $F\colon \mathbb{C}\to \mathbb{C}$ be defined as $$F(s) = \frac{1}{4+3s+\sqrt{s(4+s)}}.$$ My question is what is the inverse Laplace transform ...
28 views

### The sum $\sum_{i=1}^n \frac{x^i}{\Gamma(\frac{i}{\sqrt{n}})}$? [on hold]

I'm interested in evaluating the following sum : $\displaystyle{\sum_{i=1}^n \frac{x^i}{\Gamma(\frac{i}{\sqrt{n}})}}$ where $x>0$. The existence of a closed form would be great but is perhaps too ...
32 views

### Problem on series expansion and Bessel functions

One way to define Bessel functions is $$e^{\frac{x}{2}(t-\frac{1}{t})}=\sum_{n=-\infty}^{+\infty}J_{n}(x)t^n.$$ How do I prove that? I can't see a way of writing the L.H.S. as a geometrical (or ...
11 views

### Mittag Leffler Stability [closed]

How to prove "Mittag-Leffler stability implies asymptotical stability" ? What should be done? Thanks.
54 views

26 views

### 2D Cauchy Distribution Peak [closed]

Is the general form of a 2D Cauchy Peak, if A is the amplitude: $$\frac{A}{1+\frac{(x-x_0)^2}{\gamma_x^2}+\frac{(y-y_0)^2}{\gamma_y^2}}$$ $?$
26 views

### Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$\theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) ,$$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
28 views

38 views

61 views

57 views

### Integral of incomplete gamma function and limit of hypergeometric function

Let $a > 0$ and consider the integral $$\int_x^\infty \frac{\Gamma(a,t)}{t}\,dt$$ where $\Gamma(a,t)$ is the upper incomplete gamma function $$\Gamma(a,t) = \int_t^\infty x^{a-1} e^{-x} \, dx.$$ ...
136 views

### Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

The Dedekind eta function is denoted by $\eta(\tau)$, and is defined on the upper half-plane ($\Im \tau >0$). Put $\tau = i x$ where $x$ is a positive real number. The function has the following ...
31 views

### The Bessel function and finding expression

The Bessel Function $J_v$ of the first kind of order $v$ can be defined by the series expresion $$J_v(x)=\sum_{n=0} ^{\infty} \frac{(-1)^n}{n!\Gamma{(1+v+n)}}\left(\frac{x}{2}\right)^{2n+v}$$ (i) if ...
53 views

### What is the motivation behind the Bessel function of second kind

I am studying Bessel function and found the good reference by G.N. Watson At some point in page 58 he introduces the following expression due to Hankel: \begin{eqnarray} \lim_{\nu \to n} \frac{J_{\...
67 views

56 views

### How to prove that $e^{-\gamma}=\prod_{n=1}^\infty\left(1+\frac1n\right)e^{-1/n}$

Suppose we defined the Gamma function $$\frac1{\Gamma(z)}=ze^{\gamma z}\prod_{n=1}^\infty\left(1+\frac zn\right)e^{-z/n}$$ where $\gamma$ is just a constant. I want to prove that $\Gamma(1)=1$, so I ...