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

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5
votes
1answer
214 views

Integral representation of cosecant function

According to Wolfram website http://functions.wolfram.com/ElementaryFunctions/Csc/introductions/Csc/05/, There exists a "well-known" integral representation for the cosecant function, i.e. $$\csc(z):...
2
votes
2answers
609 views

Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
3
votes
2answers
240 views

Integrals of Hermite polynomials over $(-\infty, 0)$

Does there exist a simple expression for integrals of the form, $I = \int_{-\infty}^0 H_n(u) H_m(u)\, \mathrm{e}^{-u^2}\,du$, where $m$ and $n$ are nonnegative integers and $H_n$ is the $n$'th (...
4
votes
1answer
165 views

Integral using residue theorem (maybe)

I came across the following integral in a book (Kato's Perturbation Theory for Linear Operators, $\S$3.5): $\int_{-\infty}^\infty (a^2+x^2)^{-n/2}\,dx$ where $n$ is a non-negative integer and $a$ is ...
19
votes
0answers
543 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = \sum_{n=...
1
vote
1answer
489 views

Solution to equations involving Plasma dispersion function

I am trying to solve an equation involving a complex argument for the plasma dispersion function as: $z = x + \iota y$, $ x = \omega / \sqrt2 k v_{Ti} $ $ y = \nu_i /\sqrt{2} k v_{Ti} $ $S[z] = -\...
5
votes
0answers
535 views

How to find the inverse Fourier Transform of the product of two bessel functions of the first kind and a complex exponential function?

I am attempting to find a closed form or symbolic expression of the inverse Fourier transform of the product of two Bessel functions of the first kind and a complex exponential, e.g. $P(t) = IFT_w \{ ...
3
votes
2answers
90 views

Definite Integral Arising from a Double Integral

I gave an integral to a student. She reported back to me that she could not do it. I've tried a couple of approaches and have failed. I imagine it is fairly easy. It's a double integral. And, no ...
0
votes
1answer
123 views

Approximation of the Fourier Transform of General Functions in a Box

I'm trying to get a general approach for the Fourier Transform of functions $f$, only in a restricted area $-\frac M2\le x \le \frac M2$, where ${\frak F}_{f(x)}(\omega)$ exists. My idea was the ...
2
votes
1answer
88 views

Why is $\lim_{x \to 0} {\rm li}(n^x)-{\rm li}(2^x)=\log\left(\frac{\log(n)}{\log(2)}\right)$?

I'm trying to give at least some partial answers for one of my own questions (this one). There the following arose: $\hskip1.7in$ Why is $\lim_{x \to 0} {\rm li}(n^x)-{\rm li}(2^x)=\log\left(\frac{...
1
vote
1answer
357 views

Equation containing modified bessel functions and exponential function

I'm trying to find a approximation solution for the following equation: ${e^{ - x}}\left[ {{I_o}\left( x \right) + {I_1}\left( x \right)} \right] = C$ where $I_0$ and $I_1$ is the modified Bessel ...
2
votes
1answer
239 views

Convergence of the Fourier Transform of the Prime $\zeta$ Functions

I think I found a way to write the truncated Prime $\zeta$ function like this: $$ P_x(s)=\sum_{p<x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n} \sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}} \...
0
votes
0answers
140 views

Elliptical Integrals and graphing plot

I'm trying to computer plot the graphs of sn(u), cn(u) and dn(u) for k = 1/4, 1/2, 3/4, 0.9 and 0.99 And I am trying to plot 3D graphs of sn, cn and dn as functions of u and k. Here's what I have: ...
2
votes
1answer
262 views

Why do Bessel functions of the first kind come up in the following 2-dimensional Fourier transform?

The following equation is given for a function $\gamma$: $\gamma = \pm \delta \left[\int \frac{d^2 p}{(2\pi)^2}qp(1-\hat{q}\cdot \hat{p})^2 \pi a^2 e^{-|q-p|^2 a^2/4}\right]^{1/2}$ where q and p are ...
2
votes
0answers
294 views

Complete Elliptic Integral of the 3rd Kind - Residual Computation

Let us consider the following function $f(a,k)$ in the interval $a,k\in (0,1]$ : $$f(a,k)=\frac{2 \sqrt{1-a^2} \sqrt{a^2-k^2}}{\sqrt{a^2}}\Pi\left(a^2,k^2\right)$$ where $\Pi\left(a^2,k^2\right)$ is ...
10
votes
0answers
479 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found $$ P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n} \sum_{z\in\{...
4
votes
2answers
2k views

Integral of Hermite polynomial multiplied by $\exp(-x^2/2)$

What is the value of $\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}H_n(x)dx$ where $H_n(x)$ is the $n^{\small\mbox{th}}$ Hermite Polynomial (physicist's convention)?
6
votes
0answers
220 views

Hints/Help studying an Abel Differential Equation

I want to know more than qualitative information about the Abel differential equation $\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$ Since I don´t know how to solve this and as far as could see, this ...
8
votes
1answer
866 views

How to evaluate the following integral using hypergeometric function?

May I know how this integral was evaluated using hypergeometric function? $$\int \sin^n x\ dx$$ Wolframalpha showed this result but with no steps Thanks in advance.
4
votes
2answers
195 views

Convergence radius of $\log(\Gamma(\exp(x)))$?

In the context of iteration of functions I'm working with the power series for $$ \small f(x)=\log(\Gamma(\exp(x))) =\sum_{k=1}^\infty a_k x^k \sim -0.577216 x + 0.533859 x^2 + 0.325579 x^3 + 0....
8
votes
1answer
211 views

Compute $\sum_{m>n=1}^{\infty} \frac{1}{m!n!}$

Compute the series $$\sum_{m>n=1}^{\infty} \frac{1}{m!n!}$$
0
votes
1answer
397 views

How to solve $(m_{(t)} x')' + kx = 0$ Sturm Liouville equation with bessel functions

I have been working on this problem for a while now and think I need assistance. I am trying to solve with respect to $x_{(t)}$ over the interval $t = [0, \infty]$: $$(m_{(t)} x')' + kx = 0$$ $$m_{(t)...
5
votes
1answer
286 views

Are these numbers $h_{r,s}$ irrational?

I came across these numbers in my work some time ago. This type of expressions do not exist in closed form (not to confuse with Vandermonde convolution), I already know that. To simplify I denote $$P(...
3
votes
1answer
121 views

Double Integral involving modifed bessel function

I'm try to derive a closed form of the following double integral: $\int\limits_0^x {\int\limits_0^x {{e^{ - {K_1}uv}}{I_0}\left( {2{K_1}\sqrt {uv} } \right)du} dv}$; where $K_1$ is a constant. Do you ...
4
votes
1answer
167 views

Definite integral with $\mathrm{Si}$ in integrand

Does the function $$f(t) = \int_0^{\sqrt{3}} (x^2-1) \;\mathrm{Si}((x^2-1)\,t)\; \mathrm{d}x$$ have a representation in terms of elementary functions of $t$ for real, positive $t$? Here, $\mathrm{Si}...
1
vote
1answer
162 views

How to approximate $\text{li}(z)$ numerically?

I'm trying to implement a function to calculate $\pi(x)$ via Riemann's formula: $$ \pi(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\...
2
votes
1answer
923 views

Electric Potential of an off axis charge (Legendre Generating Function)

An insulated disk, uniform surface charge density $\sigma$, of radius R is laid on the xy plane. Deduce the electric potential $V(z)$ along the z-axis. Next ...
2
votes
1answer
1k views

Normalization of the Bessel function

I would greatly appreciate assistance with the following problem. show: $$\int _0 ^\infty J_n(x)dx = 1; \forall n \in \mathbb{N}^+$$ for $J_o,$ use $$\mathscr{L}{J_o(at)} = \int _0 ^\infty e^{-pt}...
1
vote
1answer
230 views

Fractional part, periodic function

I don't know how to solve this problem: Let $f$ be a continuous real function such that $\{f(x)\} = f(\{x\})$ for each $x$ ($\{x\}$ is the fractional part of number x) Prove that then $f$ or $f(x)-...
3
votes
3answers
114 views

Can this function be expressed in terms of other well-known functions?

Consider the function $$f(a) = \int^1_0 \frac {t-1}{t^a-1}dt$$ Can this function be expressed in terms of 'well-known' functions for integer values of $a$? I know that it can be relatively simply ...
9
votes
1answer
538 views

Evaluation of an integral involving the Lambert W function

Wikipedia claims that $$\int_0^\infty W\left(\frac{1}{x^2}\right) \,\text dx=\sqrt{2\pi}$$ and a numerical computation seems to confirm this. How can this result be proven?
1
vote
1answer
322 views

asymptotic behavior of the real part of the Riemann zeta function for $0<\sigma<1$

consider the zeta function $\zeta(\sigma+it)$ for $\sigma>1$ : $$\zeta(\sigma+it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma+it}}$$ And: $$\zeta(\sigma-it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma-it}}$$ ...
0
votes
1answer
42 views

Why is this function homogenous to the specified degree?

I have this function $$ w(q) = (1 - \alpha)q^nBk^\alpha + c $$ The paper I'm reading says that w is homogenous of degree $$ n/(1-\alpha) $$ and so small differences in q cause large differences ...
2
votes
0answers
196 views

Problem with understanding first (and second) derivative of a two-sided infinite series

For the function $$f(x)=b^x-1 = x_1 \qquad g(x)=\log(1+x)/\log(b) $$ and its iterative notation $$ x_0=x \qquad x_h=f(x_{h-1})=g(x_{h+1}) \qquad x_{-1}=g(x_0) $$ with b from the interval $1 \lt b \lt ...
3
votes
2answers
55 views

$\frac{\mathrm{d} g(x)}{\mathrm{d}x}=h(x)$ and $\frac{\mathrm{d} h(x)}{\mathrm{d}x}=g(x)$ where $h(x)\neq g(x)$

Is there any other solution to : $$\frac{\mathrm{d} g(x)}{\mathrm{d}x}=h(x)$$ $$\frac{\mathrm{d} h(x)}{\mathrm{d}x}=g(x)$$ other than $h(x)=g(x)=e^x$? By varying $\alpha,\beta$ in $$\frac{\mathrm{d} ...
4
votes
0answers
247 views

Saddle point and stationary point approximation of the Airy equation

Happy New Year to you all. Let $$\tag 1 J(N)=\int_a^b e^{Nf(x)}dx$$ where $N\in\mathbb R$ and $N>>1$ and $f(x)$ has a global maximum at $x=x_0$ with Taylor expansion $$f(x) \approx f(x_0)-|f'...
1
vote
1answer
177 views

Is a Macdonald function a Bessel function with imaginary argument??

I mean that $$ K_{a} (x)= CJ_{a}(ix).$$ Here $C$ is a complex number, and $a$ is real. So is the Macdonald function a Bessel function in disguise (or proportional) of complex argument??
0
votes
0answers
73 views

Summing equally spaced samples of a periodic function

I'm a little stuck at the moment and wondered if someone could point me in the direction of the theory I need to read. I have a $2\pi$-periodic function, $f:\mathbb{R}\rightarrow\mathbb{R}$ which I ...
18
votes
2answers
2k views

The Gamma function and the Pi function

I have been studying differential equation, in particular special functions. Euler's Gamma function, and Gauss's Pi function are essentially the same, differing only by an offset of one unit. for $...
2
votes
1answer
97 views

About the values of the $\Gamma$ function

The $\Gamma$ function is defined by $$\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}dt$$ where $z$ is a complex number. We know that if $z$ is real then the values of $\Gamma$ are also real. I am ...
3
votes
0answers
146 views

A Thue-Morse Zeta function ( Generalized Riemann Zeta function and new GRH )

Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ a complex number. Odiuos Number Now consider the sequence of functions below $f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$ ...
6
votes
2answers
464 views

Prove that $\int_0^1 \psi{(x) \sin(2 n \pi x)} \space\mathrm{dx}=-\frac{\pi}{2}$

Prove that $$\int_0^1 \psi{(x) \sin(2 n \pi x)} \space\mathrm{dx}=-\frac{\pi}{2}, \space n\ge1$$ where $\psi(x)$ - digamma function
2
votes
1answer
1k views

error function (erf) with better precision

Currently I'm using this C++ routine to approximate the error function ...
4
votes
2answers
563 views

Solving the integral of a Modified Bessel function of the second kind

I would like to find the answer for the following integral $$\int x\ln(x)K_0(x) dx $$ where $K_0(x)$ is the modified Bessel function of the second kind and $\ln(x)$ is the natural-log. Do you have ...
7
votes
1answer
179 views

Finding x in $\frac{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,1-x)}{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,x)} = \sqrt{n}$

I was trying to find a closed-form for $0<x<1$ in, $$\frac{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,1-x)}{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,x)} = \sqrt{n}$$ where $\,_2F_1(a,b,c,z)$ ...
2
votes
0answers
76 views

asymptotics of $ J_{iu} (ia)$ for a Bessel function

Let $J_{iu}(ia)$ be the Bessel function of imaginary order. ($a$ is a real number (positive or negative) and $u$ is also real.) In the limit $u \to \infty $ how does the function $J_{iu} (ia)$ behave?...
3
votes
2answers
68 views

Showing integrability (Riemann)

I was trying to show whether or not the function: $f: [0,1 ] \rightarrow \mathbb{R}$ $f(x)= \frac {1}{n}$ for $x = \frac {1}{n}$ $(n \in \mathbb{N})$ and $f(x) = 1$ if the condition isn't ...
2
votes
1answer
79 views

What is the “name” of this function?

There is a function I met in complex analysis. $$f(\lambda) = \int \limits_{-\infty}^{\infty}\frac{e^{i\lambda x}}{\sqrt{1 + x^{2n}}}dx$$
10
votes
2answers
794 views

An inequality from the handbook of mathematical functions (by Abramowitz and Stegun)

Prove that $$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$
0
votes
1answer
497 views

Identity concerning $e^{ia\sin{x}}$ as a series of bessel functions

Prove the following identity: \begin{equation} e^{ia\sin{x}}=\sum_{-\infty}^{+\infty}J_k(a) e^{ikx}, \end{equation} where $a$ is a real constant and $J_k$ is the Bessel function of the first type of ...