Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).
15
votes
1answer
311 views
Intuition why the volume and surface area of the unit sphere eventually decrease
The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$
and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$
both attain maximum values for finite $n$. We can ...
2
votes
2answers
95 views
Dirac delta in polar coordinates
Given
$$x=r\,\cos\theta\\y=r\,\sin\theta$$ and
$$x'=r'\,\cos\theta'\\y'=r'\,\sin\theta'$$
how can I express
$$\delta(x'-x)\delta(y'-y)$$ in terms of the polar coordinates?
And the more general ...
6
votes
1answer
49 views
Need help with $\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx$
I need help with solving this integral:
$$\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx,$$
where $\text{Li}_{s}(z)$ is the polylogarithm.
26
votes
0answers
246 views
+500
Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$
Consider the following integral:
$$\mathcal{I}(\mu,\nu)=\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx,$$
where $J_\mu(x)$ is the Bessel function of the first kind:
...
8
votes
3answers
83 views
How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?
I need some help with solving this integral involving Bessel function:
$\hspace{2in}\displaystyle\int_0^\infty$$J_0(x)\ $$\text{sinc}(\pi\,x)\ $$e^{-x}\,\mathrm dx.$
0
votes
1answer
88 views
Finding $\int_{-\infty}^{\infty} e^{-x^2/2}x^{2n}\,\mathrm dx$ by symmetry
I can easily show that (substituting $\frac{x^2}{2} = t $ using the identity for Gamma function of $n+\frac{1}{2}$, then further expanding $\Gamma(n+\frac{1}{2})=\dfrac{(2n-1)!! \sqrt{\pi}}{2^n}$ and ...
1
vote
1answer
99 views
$ \sum_{y=1}^\infty {}_1F_1(1-y;2;-\pi\lambda c) \frac{\lambda^y}{y!} $
I am not able to solve the following sum. Can you please provide any hints ?
$$ \sum_{y=1}^\infty {}_1F_1(1-y;2;-\pi\lambda c) \frac{\lambda^y}{y!} $$
Note that the 3rd parameter of the Confluent ...
0
votes
0answers
18 views
Fourier transform of $f(r,r',\theta,\theta')$
How can I calculate the FT of:
$$\sum_{n=-\infty}^{\infty}\,e^{in(\theta' -\theta)}\,f_n(r,r')=\sum_{n=-\infty}^{\infty}\,e^{in(\theta' -\theta)}\,\frac{J_n(\alpha r)J_n(\alpha r')}{[(\alpha ...
3
votes
0answers
118 views
Identity involving $\zeta(3)$
This is related to this question and my partial answer to it. I've found that the proof of the 2nd identity reduces to showing that
...
1
vote
0answers
28 views
Integral of Bessel function of the first kind and exponential function
I would need to know if there's a closed form for the following integral:
$$\int_{0}^{\infty} x^{-1}J_{\frac{1}{2}}(\pi x)J_{\frac{1}{2}}(\pi x)\exp(-b(x-x_0)^2)$$
with $b>0$ and $x_0\in ...
1
vote
1answer
39 views
Question about Lambert W function
I'm looking for a series for $W_0(x)$ for x $\in [\frac{-1}{e},\infty [$ but every time i found only for $x\in [\frac{-1}{e}, \frac{1}{e}]$
and what about a series for $W_-1(x)$
if it is no series ...
1
vote
1answer
31 views
what is exactly analytic continuation of the product log function
When I solve in wolfram equation like this $xe^x=z$
they give me the solution $x=W_n(z)$
I know about $x=W_0(z) $ and $x=W_{1}(z)$ but for $n$ I searched in the internet but I didn't find anything ...
5
votes
1answer
136 views
Tough Inverse Fourier Transform
In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
13
votes
3answers
135 views
Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$
I'm trying to figure out how to evaluate the following:
$$
J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx
$$
I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies ...
3
votes
2answers
45 views
On the Hurwitz Zeta Function
In my mathematics course in Uni. (I'm a physics student) my prof. gave us the following exercise: to express the Hurwitz Zeta function $\zeta(2k+1,\frac{1}{4})$ with $k=1,2,3,\dots$ in terms of the ...
5
votes
2answers
61 views
$k$-th moment of product of gaussian and sinc
I would like to calculate the following integrals:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(\frac{\sin(\pi a x)}{\pi ax}\right)^2\quad \exp(-bx^2)\,dx$$
$$\int_{-\infty}^{+\infty} \quad ...
1
vote
0answers
60 views
General solution for $M^{\circ -1 }(y)=x $ when $g(x)e^{f(x) }=y$
Reading this question $e^{C/x }-1=D/(x + a) $, i found my self completely unable to do anything. This is much more hard for me than my easy exercises about Lambert $W$-function.
So I probably need ...
1
vote
1answer
27 views
Recurrence inequality for Dirichlet's eta function.
I'm studying the following function:
$\theta(p)=\eta(p)\eta(p-2)-\frac{p-1}{p}\eta^2(p-1)$,
where $\eta$ - Dirichlet's eta function. If we build a plot for $p\in [1,150]$, it's easy to see that it's ...
1
vote
2answers
67 views
Summation involving subfactorial function
Inspired by this post:
Does the following series converge; if so, to what value does it converge? $$ \sum_{n = 2}^\infty \left|\frac{!n \cdot e}{n!} - 1\right|$$ I am looking for a closed form for ...
0
votes
0answers
36 views
How to solve for $z$ in $\dfrac{xy}{1-x}=(1-z)(x-x^{1/z})$
How to solve the following for $z$:
$$\frac{xy}{1-x}=(1-z)(x-x^{1/z})$$
where $0 < x < 1$, $\;0 < y < 1$, $\;0 < z \leq 1$.
0
votes
0answers
23 views
how we can find the continued fraction of incomplete Gamma function
the continued fraction is a beautiful mathematical tool
when i read about incomplete Gamma function in wikipedia I saw the continued fraction of it
i have some information about find the continued ...
0
votes
1answer
32 views
Simplify square of sinc functions
I need to simplify if possible the following:
$$\left(i^n\cdot \operatorname{sinc}\big(\pi(x-\tfrac{n}{2})\big)+(-i)^n\cdot \operatorname{sinc}\big(\pi(x+\tfrac{n}{2})\big)\right)^2$$
with $n \in ...
1
vote
0answers
19 views
Simplification of Kampé de Fériet function
I was dealing with a convolution type integral
$$
\int^z_0 t^m {}_0F_1(;1;-t) \: {}_2F_3\Big( 1,1;2,m,m+1 ; -a t\Big) \:\mathrm{d}t
$$
By applying one of the identities in Exton's book, the solution ...
1
vote
0answers
19 views
Closed form for $k$-th moment
I would like to calculate this $k$-th moment:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(i^n\frac{\sin(\pi a x+\frac{n\pi}{2})}{\pi ax+\frac{n\pi}{2}}+(-i)^n\frac{\sin(\pi a ...
3
votes
2answers
69 views
What is the functional inverse (with respect to $h$ (!)) of $f^{\circ h}(x)={F(h) +x F(h-1) \over F(1+h) +x F(h) }$?
I've considered the fractional iteration of $f(x) = {1 \over 1+x} $ for which the general expression depending on the iteration-height parameter $h$ might be assumed by the formula
$$ f^{\circ h}(x) ...
2
votes
1answer
59 views
Fourier Transforms of shifted sinc funtions
I would like to calculate the Fourier transform of the following functions:
$$\left(\dfrac{\sin(\pi x\pm\pi n/2)}{\pi x\pm\pi n/2}\right)^2$$
$$\dfrac{\sin(\pi x+\pi n/2)}{\pi x+\pi ...
0
votes
0answers
39 views
How to graph the equation of a special mathematical function like $y=\zeta (x)$
I can't find any way to graph any special mathematical function like $y=\zeta (x)$, $y=\Gamma (x)$, etc.
Because of that, I ask this question to learn about examples, to be able to graph other ...
6
votes
1answer
175 views
Can it be shown that $Y_0(\lambda_n a)J_0(\lambda_n a) - J_0(\lambda_n a)Y_0(\lambda_n a) \ne 0$?
Background
I am currently looking into the task of describing a transient temperature field $\theta(r,t)$ across the thickness $a \leq r \leq b$ of an infinitely long and hollow cylinder exposed to a ...
1
vote
0answers
23 views
question about Bernoulli number
we know that we can generate the Bernoulli number using this equation
$(1+B)^n=B^{[n]}$
but how we can prove it ?
please help
and thanks for all
0
votes
1answer
22 views
Ordinary generating function for Bernoulli polynomial
I know the exponential generating function for the Bernoulli polynomial $B_n(x)$:$$\frac{te^{tx}}{e^t-1}=\sum_{n=0}^\infty B_n(x)\frac{t^n}{n!}.$$
But is there an ordinary generating function? i.e a ...
6
votes
2answers
86 views
Computation of $\int_0^{\pi} \frac{\sin^n \theta}{(1+x^2-2x \cdot \cos \theta)^{\frac{n}{2}}} \, d\theta$
Show that
$$\begin{align*} \forall x \in [-1,1]: \int_0^{\pi} \frac{\sin^n \theta}{(1+x^2-2x \cdot \cos \theta)^{\frac{n}{2}}} \, d\theta &= c_n \tag{1} \\ \int_0^{\pi} \frac{\sin^{n+2} ...
1
vote
1answer
34 views
$0$-th moment of product of gaussian and sinc function
I would like to calculate the following integrals:
$$\int_{-\infty}^{+\infty} \quad \left(\frac{\sin(\pi a x)}{\pi ax}\right)^2\quad \exp(-bx^2)\,dx$$
$$\int_{-\infty}^{+\infty} \quad ...
2
votes
0answers
54 views
Convexity of polylogarithms
I want to prove the following proposition:
The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$.
And, as I think, the same is true for the function $w\to (-Li_{p}(-e^w))^{1/p}$ for ...
14
votes
1answer
168 views
$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$
I need help with calculating this integral:
$$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$
where ...
16
votes
4answers
195 views
$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$
Please help me to solve this integral:
$$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx.$$
I managed to calculate an indefinite integral of the left part:
$$\int\frac{\cos x}{\sin ...
1
vote
2answers
51 views
Convolution of $1/(1+x^2)$ and $\exp(-x^2/(4t))$
Is there a closed form formula for the convolution of $1/(1+x^2)$ and $\exp(-x^2/(4t))$, where $t>0$, i.e. the integral
$$\int_{-\infty}^\infty ...
2
votes
1answer
47 views
Integral of product of Bessel functions of the first kind
I would like to solve the integral:
$$\int_0^{+\infty}\quad rJ_n(ar)J_n(br)\quad dr$$
Is there any closed form for it?
Thanks!
0
votes
0answers
26 views
Square equivalent of $circ(r)$
I would like to know if there is a similar function to
$$circ(\sqrt{x^2+y^2})=1 , 0\leq \sqrt{x^2+y^2}\leq 1$$
but with a square domain $0\leq x\leq 1$ and $0\leq y\leq 1$.
If yes, which is its ...
2
votes
3answers
70 views
What type of Hypergeometric series is this?
I am trying to find a closed form for the series
$$ \sum^\infty_{n=0} \frac{1}{n!} \frac{1}{n+1}(-z)^n {}_2F_2\left(m,n+1;\frac{1}{2},n+2; b z\right)$$
$m$ is a nonzero positive integer, and $b$, ...
2
votes
1answer
50 views
Integral identity with square of Jacobi polynomial
This has stumped me for a while: I have a function $\zeta_k^S(x)$ that can be expressed using Jacobi polynomials $P_k^{(\alpha,\beta)}(x)$:
...
3
votes
1answer
66 views
Continued fraction expansion related to exponential generating function
A recent SciComp.SE Question motivates us to ask for a nice continued fraction expansion of the following Maclaurin series:
$$ f(x) = \sum_{n=0}^\infty \frac{B_n\; x^{n+3}}{n! (n+3)} = \int_0^x ...
1
vote
1answer
25 views
Difference between Rician distribution and Gaussian distribution
could any one please tell me the difference between Rician and Gaussian Distribution and the advantages of using one over other please.With some mathematical proof would be truly appreciated
Thank ...
3
votes
1answer
44 views
$\int_0^1 \Bigl(\mathrm{Li}_2\bigl(\frac{x-1}{x}\bigr)\Bigr)^2 \mathrm{d}x$
This question has some relationship to this integral: Let $\mathrm{Li}_2$ be the dilogarithm. Then, numerically,
$$
\int_0^1 \Bigl(\mathrm{Li}_2\bigl(\frac{x-1}{x}\bigr)\Bigr)^2 \mathrm{d}x
=
...
8
votes
1answer
90 views
$\int_0^\infty\text{Ci}(x)^3\mathrm dx$
Is there a closed form for this integral:
$$\int_0^\infty\text{Ci}(x)^3\mathrm dx,$$
where $\text{Ci}(x)=-\int_x^\infty\frac{\cos z}{z}\mathrm dz$ is the cosine integral?
3
votes
0answers
44 views
The Tribonacci constant and the Dragon
Let $x = \frac{\ln T}{\ln 2} = 0.879146\dots$ where $T$ is the tribonacci constant, then x solves the transcendental equation,
$$4^x(2^x-1)=(2^x+1)$$
Let $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
0
votes
1answer
21 views
Show that $\int_0^t\!\!\left(t^2-x^2\right)^n\mathbb{d}x=\frac{\sqrt{\pi}}{2}t^{2n+1}\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}$
The question asks to prove the identity:
$$\int_0^t\!\!\left(t^2-x^2\right)^n\mathbb{d}x=\frac{\sqrt{\pi}}{2}t^{2n+1}\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}$$
where $n\in\mathbb{Z}$
I have no idea ...
1
vote
1answer
43 views
Find local maxima of this quadratic function
How can I find local maxima of this quadratic function?
$$f(x) = \sum _{i=1}^n -\frac{(z_i - x)_+^2}{2} - \left\{((\frac{(z_i - x)_+^2}{2})-(\frac{(y_i - x)_+^2}{2}) ) * c_i\right\} $$
which ...
1
vote
0answers
49 views
Integral involving Gauss Hypergeometric function, power, exponential and Bessel Function
I am trying to evaluate the following integral involving the Gauss Hypergeometric function, power, exponential and a Bessel Function:
$$
\int_0^\infty x e^{-cx^2} {_2F_1(1,\frac{2} {ab},1+\frac{2} ...
0
votes
0answers
57 views
Integral involving exponential, power and Bessel function
Is there any formula for calculating the following definite integral, including exponential and Bessel function?
$$ \int_0^{a}x^{-1} e^{x}I_2(bx)dx$$
Thanks in advance
5
votes
1answer
89 views
A problematic integral: $\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$
Is there a special trick to calculate this integral?
$$\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$$
for $\lambda>0$.
