Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).
3
votes
1answer
41 views
Meaning of Euler Reflection Formula
A book that I am reading says that Euler reflection formula $$\Gamma (x)\Gamma (1-x)=\frac{\pi}{\sin \pi x}$$, in a sense, shows that $1/\Gamma(x)$ is half of the sine function.
Does anyone ...
5
votes
2answers
104 views
How to prove a generalized integral identity
$$
\int_{0}^{\infty }\frac{t}{(e^{2\pi t}-1)(1+t^{2})}dt=-\frac{1}{4}+\frac{\gamma}{2}
$$
where $\gamma$ = Euler Gamma
$$
\int_{0}^{\infty }\frac{t}{( e^{2\pi t}-1)(1+t^{2}) ^{2}}dt=\frac{\pi^2}{24} ...
3
votes
1answer
65 views
Improper integral and special functions
I'd like to have an expression of the following integral:
$$\int_0^{+\infty} \left(\sqrt{1+x^4} - x^2\right) dx$$
in terms of some special functions (but not in the form given by Wolfram Alpha).
1
vote
4answers
78 views
Is $\sum^{n-1}_{k=1}{n\choose k}x^{n-k}y^k$ always even?
Is
$$ f(n,x,y)=\sum^{n-1}_{k=1}{n\choose k}x^{n-k}y^k,\qquad\qquad\forall~n>0~\text{and}~x,y\in\mathbb{Z}$$
always divisible by $2$?
3
votes
0answers
47 views
An inequality with Gamma Function
Consider the following function for any $a, b > 0$
$$
\
g\left( a,b\right) = \frac{%
3\Gamma \left( 3b+1\right) }{\Gamma \left( \frac{1}{a}+3b+1\right) }-\frac{%
5\Gamma \left( 2b+1\right) ...
6
votes
2answers
102 views
How to prove $4\times{_2F_1}(-1/4,3/4;7/4;(2-\sqrt3)/4)-{_2F_1}(3/4,3/4;7/4;(2-\sqrt3)/4)\stackrel?=\frac{3\sqrt[4]{2+\sqrt3}}{\sqrt2}$
I have the following conjecture, which is supported by numerical calculations up to at least $10^5$ decimal digits:
...
8
votes
1answer
58 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.
2
votes
2answers
121 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 ...
0
votes
0answers
20 views
Fourier transform of $f(r,r',\theta,\theta')$
How can I calculate the FT of:
$$\sum_{n=-\infty}^{\infty}\,e^{in(\theta' -\theta)}\,f_n(r,r')=\sum_{n=-\infty}^{\infty}\,e^{in(\theta' -\theta)}\,\frac{J_n(\alpha r)J_n(\alpha r')}{[(\alpha ...
1
vote
0answers
30 views
Integral of Bessel function of the first kind and exponential function
I would need to know if there's a closed form for the following integral:
$$\int_{0}^{\infty} x^{-1}J_{\frac{1}{2}}(\pi x)J_{\frac{1}{2}}(\pi x)\exp(-b(x-x_0)^2)$$
with $b>0$ and $x_0\in ...
1
vote
1answer
33 views
what is exactly analytic continuation of the product log function
When I solve in wolfram equation like this $xe^x=z$
they give me the solution $x=W_n(z)$
I know about $x=W_0(z) $ and $x=W_{1}(z)$ but for $n$ I searched in the internet but I didn't find anything ...
1
vote
1answer
42 views
Question about Lambert W function
I'm looking for a series for $W_0(x)$ for x $\in [\frac{-1}{e},\infty [$ but every time i found only for $x\in [\frac{-1}{e}, \frac{1}{e}]$
and what about a series for $W_-1(x)$
if it is no series ...
0
votes
0answers
36 views
How to solve for $z$ in $\dfrac{xy}{1-x}=(1-z)(x-x^{1/z})$
How to solve the following for $z$:
$$\frac{xy}{1-x}=(1-z)(x-x^{1/z})$$
where $0 < x < 1$, $\;0 < y < 1$, $\;0 < z \leq 1$.
10
votes
3answers
112 views
How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?
I need some help with solving this integral involving Bessel function:
$\hspace{2in}\displaystyle\int_0^\infty$$J_0(x)\ $$\text{sinc}(\pi\,x)\ $$e^{-x}\,\mathrm dx.$
1
vote
0answers
61 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 ...
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 ...
13
votes
3answers
142 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 ...
1
vote
0answers
20 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:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(i^n\frac{\sin(\pi a x+\frac{n\pi}{2})}{\pi ax+\frac{n\pi}{2}}+(-i)^n\frac{\sin(\pi a ...
1
vote
1answer
110 views
$ \sum_{y=1}^\infty {}_1F_1(1-y;2;-\pi\lambda c) \frac{\lambda^y}{y!} $
I am not able to solve the following sum. Can you please provide any hints ?
$$ \sum_{y=1}^\infty {}_1F_1(1-y;2;-\pi\lambda c) \frac{\lambda^y}{y!} $$
Note that the 3rd parameter of the Confluent ...
0
votes
1answer
32 views
Simplify square of sinc functions
I need to simplify if possible the following:
$$\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$$
with $n \in ...
0
votes
0answers
40 views
How to graph the equation of a special mathematical function like $y=\zeta (x)$
I can't find any way to graph any special mathematical function like $y=\zeta (x)$, $y=\Gamma (x)$, etc.
Because of that, I ask this question to learn about examples, to be able to graph other ...
1
vote
2answers
73 views
Summation involving subfactorial function
Inspired by this post:
Does the following series converge; if so, to what value does it converge? $$ \sum_{n = 2}^\infty \left|\frac{!n \cdot e}{n!} - 1\right|$$ I am looking for a closed form for ...
3
votes
2answers
70 views
What is the functional inverse (with respect to $h$ (!)) of $f^{\circ h}(x)={F(h) +x F(h-1) \over F(1+h) +x F(h) }$?
I've considered the fractional iteration of $f(x) = {1 \over 1+x} $ for which the general expression depending on the iteration-height parameter $h$ might be assumed by the formula
$$ f^{\circ h}(x) ...
0
votes
0answers
32 views
question about Bernoulli number
we know that we can generate the Bernoulli number using this equation
$(1+B)^n=B^{[n]}$
where $B_n$ is Bernoulli number
but how we can prove it ?
is there any help
thanks for all
34
votes
0answers
410 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:
...
1
vote
1answer
34 views
$0$-th moment of product of gaussian and sinc function
I would like to calculate the following integrals:
$$\int_{-\infty}^{+\infty} \quad \left(\frac{\sin(\pi a x)}{\pi ax}\right)^2\quad \exp(-bx^2)\,dx$$
$$\int_{-\infty}^{+\infty} \quad ...
6
votes
2answers
103 views
Computation of $\int_0^{\pi} \frac{\sin^n \theta}{(1+x^2-2x \cdot \cos \theta)^{\frac{n}{2}}} \, d\theta$
Show that
$$\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} ...
5
votes
2answers
61 views
$k$-th moment of product of gaussian and sinc
I would like to calculate the following integrals:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(\frac{\sin(\pi a x)}{\pi ax}\right)^2\quad \exp(-bx^2)\,dx$$
$$\int_{-\infty}^{+\infty} \quad ...
2
votes
1answer
59 views
Fourier Transforms of shifted sinc funtions
I would like to calculate the Fourier transform of the following functions:
$$\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 ...
1
vote
2answers
53 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
$$\int_{-\infty}^\infty ...
2
votes
1answer
48 views
Integral of product of Bessel functions of the first kind
I would like to solve the integral:
$$\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$$
but with a square domain $0\leq x\leq 1$ and $0\leq y\leq 1$.
If yes, which is its ...
1
vote
1answer
27 views
Recurrence inequality for Dirichlet's eta function.
I'm studying the following function:
$\theta(p)=\eta(p)\eta(p-2)-\frac{p-1}{p}\eta^2(p-1)$,
where $\eta$ - Dirichlet's eta function. If we build a plot for $p\in [1,150]$, it's easy to see that it's ...
2
votes
3answers
71 views
What type of Hypergeometric series is this?
I am trying to find a closed form for the series
$$ \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)$$
$m$ is a nonzero positive integer, and $b$, ...
1
vote
1answer
26 views
Difference between Rician distribution and Gaussian distribution
could any one please tell me the difference between Rician and Gaussian Distribution and the advantages of using one over other please.With some mathematical proof would be truly appreciated
Thank ...
2
votes
1answer
52 views
Integral identity with square of Jacobi polynomial
This has stumped me for a while: I have a function $\zeta_k^S(x)$ that can be expressed using Jacobi polynomials $P_k^{(\alpha,\beta)}(x)$:
...
3
votes
0answers
45 views
The Tribonacci constant and the Dragon
Let $x = \frac{\ln T}{\ln 2} = 0.879146\dots$ where $T$ is the tribonacci constant, then x solves the transcendental equation,
$$4^x(2^x-1)=(2^x+1)$$
Let $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
0
votes
1answer
24 views
Show that $\int_0^t\!\!\left(t^2-x^2\right)^n\mathbb{d}x=\frac{\sqrt{\pi}}{2}t^{2n+1}\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}$
The question asks to prove the identity:
$$\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})}$$
where $n\in\mathbb{Z}$
I have no idea ...
3
votes
1answer
45 views
$\int_0^1 \Bigl(\mathrm{Li}_2\bigl(\frac{x-1}{x}\bigr)\Bigr)^2 \mathrm{d}x$
This question has some relationship to this integral: Let $\mathrm{Li}_2$ be the dilogarithm. Then, numerically,
$$
\int_0^1 \Bigl(\mathrm{Li}_2\bigl(\frac{x-1}{x}\bigr)\Bigr)^2 \mathrm{d}x
=
...
9
votes
1answer
95 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?
16
votes
4answers
207 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:
$$\int\frac{\cos x}{\sin ...
1
vote
1answer
43 views
Find local maxima of this quadratic function
How can I find local maxima of this quadratic function?
$$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\} $$
which ...
1
vote
0answers
51 views
Integral involving Gauss Hypergeometric function, power, exponential and Bessel Function
I am trying to evaluate the following integral involving the Gauss Hypergeometric function, power, exponential and a Bessel Function:
$$
\int_0^\infty x e^{-cx^2} {_2F_1(1,\frac{2} {ab},1+\frac{2} ...
0
votes
0answers
63 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
91 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 $\lambda>0$.
2
votes
1answer
37 views
Hermite's equation of order $\alpha$
Show that the general solution of Hermite's equation of order $\alpha$:
$${y}''-2x{y}'+2\alpha y=0$$
$$is$$
$$y(x)=c_{0}y_{1}(x)+c_{1}y_{2}(x)$$
where $y_{1}(x)$ and $y_{2}(x)$ are power series ...
1
vote
0answers
14 views
Contiguous hypergeometric functions
I'm looking for relation, if there exists of course, simplifying the following sum:
$$F(\alpha,\beta;\gamma_1;x)+F(\alpha,\beta;\gamma_2;x)$$ where $F$ denotes the Gauss hypergeometric function, ...
3
votes
2answers
47 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 ...
17
votes
1answer
181 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 ...
