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

learn more… | top users | synonyms

2
votes
1answer
480 views

Verification of integral over $\exp(\cos x + \sin x)$

I found the following integral in a paper I was reading: \begin{equation} \frac{1}{2\pi} \int\limits_{-\pi}^{\pi} \exp\left(a \cos x + b \sin x\right) dx = I_0\left(\sqrt{a^2+b^2}\right), \end{...
2
votes
2answers
115 views

How to solve $x \geq \frac{y}{z-\ln{x}}$ for positive variables?

How can you solve $x \geq \frac{y}{z-\ln{x}}$ for $x$ when the variables are real positive values? I am only really interested in the case where the values are large and $z > \ln x$. How ...
0
votes
2answers
184 views

Bessel function confirmation

I'm trying to see that $J_0(x)$ is indeed a solution for the Bessel equation $x^2y''+xy'+x^2y=0$, so: $$J_0(x)=\sum_{k=0}^\infty \frac{(-1)^kx^{2k}}{(k!)^22^{2k}}$$ Pluging it in the equation and and ...
2
votes
1answer
114 views

Constant term of recursively defined polynomials related to the Lambert W function

The Lambert $W$ function has the property that $$ W'(x) = \frac{W(x)}{x[1+W(x)]}, $$ and using this one can show that its Taylor expansion about $x=a$ has the form $$ W(x) = W(a) + \sum_{n=1}^{\...
1
vote
0answers
70 views

The cdf of a beta variable, evaluated at the mean

Consider a Beta random variable $X$ with shape parameters $k/2$ and $(d-k)/2$, where the parameters $k, d$ are integers that satisfy $0 < k < d$. What is the best possible upper bound for the ...
11
votes
3answers
386 views

Integral $\int\limits_0^\infty \prod\limits_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx$

Does anybody know how to prove this identity? $$\int_0^\infty \prod_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx=\frac{\sqrt{\pi}}{2}\frac{\Gamma \left(a+\frac{1}{2}\right)\...
8
votes
2answers
221 views

The double integral $\int_0^1 \int_0^1 \frac{1-x}{1-xy}(-\log(xy))^s \, dx \, dy$

$$\int_0^1 \int_0^1 \frac{1-x}{1-xy}(-\log(xy))^s \, dx \, dy=\Gamma(s+2)\left(\zeta(s+2) -\frac{1}{s+1}\right) \quad \Re(s)>-2$$ How to prove this identity?
6
votes
2answers
629 views

Turning an elliptic curve over C into a complex torus

I have been reading a lot about the Weierstrass $\wp$ function and I understand the parameterization of an elliptic curve with the elliptic function( i.e. $x=\wp(z)$ and $y=\wp^\prime(z)$). I would ...
1
vote
2answers
71 views

Roots of the equation $I_1(b x) - x I_0(b x) = 0$

I'm interested in the roots of the equation: $I_1(bx) - x I_0(bx) = 0$ Where $I_n(x)$ is the modified Bessel function of the first kind and $b$ is real positive constant. More specifically, I'm ...
3
votes
2answers
789 views

A hard log definite integral: $\int_0^{\pi/4}\ln^3\sin x\,\mathrm dx$

Show that: $$\int_0^{\pi/4}\ln ^3\sin x\text{d}x=\frac34\text{Im}\left(\text{Polylog}\left(4,i\right)\right)-3\text{Im}\left(\text{Polylog}\left(4,\frac12+\frac{i}{2}\right)\right)-\frac{23}{128}\pi^3\...
11
votes
3answers
334 views

Evaluating $\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} dx$

What is the value of the following integral? $$\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} \,dx$$ Here $\Gamma(x)$ is Euler's gamma function. EDIT: Can we improve the upper bound strictly smaller than $1$? ...
2
votes
0answers
100 views

A more general exponential integral

More generalized on the previous question: A improper integral with Glaisher-Kinkelin constant Show that : $\displaystyle\int_0^\infty\frac{\text{e}^{-ax}}{x^2}\left(\frac{1}{1-\text{e}^{-x}}-\frac{1}{...
2
votes
1answer
84 views

Proof of $\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$?

Can anyone show a proof of $$\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$$ where $0<a<1$, and $$\Gamma(a)=\int_0^\infty t^{a-1}e^{-t}\,dt.$$ Thank you.
5
votes
1answer
127 views

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$ I don't know how to do this ? Note that $\gamma $ is the Euler-Mascheroni constant
4
votes
2answers
908 views

Calculate an integral involving Hermite polynomials

I have to calculate the integral $$\frac{1}{\sqrt{2^nn!}\sqrt{2^ll!}}\frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty}H_n(x)e^{-x^2+kx}H_l(x)\;\mathrm{d}x$$ where $H_n(x)$ is the $n^{th}$ Hermite ...
0
votes
1answer
173 views

Bessel function equation

I have to prove that equation: $$\tag{1} \operatorname{J}_2(xy)\ \operatorname{Y}_0 (x) - \operatorname{J}_0(x) \operatorname{Y}_2(xy) =0$$ (where $\operatorname{J}_k$ and $\operatorname{Y}_k$ are ...
2
votes
1answer
105 views

Construct a generating function for the components of a sum

Let $j \in Z_+$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Find generating function $\sum_{j}a_jx^j$ so that allows to ...
5
votes
1answer
212 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
604 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
239 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
540 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=...
5
votes
0answers
530 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
87 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
356 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
238 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
261 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
289 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
478 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
1k 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
218 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
855 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
210 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
396 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
284 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
120 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
165 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
161 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
917 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
222 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
535 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
319 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 ...