Questions tagged [special-functions]
This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).
4,518
questions
3
votes
0
answers
33
views
Closed form of dilogarithm fucntion involving many arctangents
I am trying to find closed form for this expression:
$$ - 2{\text{L}}{{\text{i}}_2}\left( {\frac{1}{3}} \right) - {\text{L}}{{\text{i}}_2}\left( {\frac{1}{6}\left( {1 + i\sqrt 2 } \right)} \right) - {\...
- 2,042
1
vote
1
answer
87
views
$\Gamma (x) \cos(ax)$ identity
I am asked to show that $$\Gamma (x) \cos(ax) = b^x \int_{0}^{\infty} \mathrm{d} t \enspace t^{x-1} e^{-bt \cos(a)} \cos(bt \sin(a)).$$
A change of variables $t \to \frac{t}{b}$ shows that $b$ is ...
- 63
0
votes
0
answers
67
views
Solving the Elliptic Integral of the First Kind with Imaginary Moduli
The Complete Elliptic Integral of the First Kind is defined as the following:
$$K(m)=\int_0^{\frac{\pi}{2}}\frac{dx}{\sqrt{1-m\sin^2(x)}}$$
Given this definition, I would like to solve the following ...
- 11
0
votes
0
answers
15
views
Bessel functions summation
I want to compute the square modulus of the following sum :
\begin{align}
\sum_p e^{i\eta p}(-\lambda)^{j-p} J_{j-p}(x)
\end{align}
Where p is an integer, j is an integer, eta is a real constant, ...
- 1
0
votes
1
answer
37
views
Struve function: simplify $\mathrm{H}_n(x) - (-1)^n \mathrm{H}_{-n}(x)$ for $n=1,2,3,...$
Consider this expression:
$$A_n(x) =\frac{\pi}{2} \left[\mathrm{\mathbf{H}}_n(x) - (-1)^n \mathrm{\mathbf{H}}_{-n}(x) \right]$$
for $n=1,2,3,...$
Where $\mathrm{\mathbf{H}}_n$ are Struve functions.
...
- 31.1k
0
votes
0
answers
47
views
Equivalence of 2 types of Airy function definition
Question
How do I show the equivalence of the following 2 definitions?
Definition 1:
$$A_1(x) :=\frac{1}{2 \pi i} \int_C \exp \left( \frac{1}{3}z^3 - xz \right) dz \tag{1}$$
Where $C$ is the path from ...
- 1
0
votes
0
answers
35
views
Discrete Schrodinger Equation - Solution as Bessel functions
Consider the DSE with no nonlinearity term: $$i\partial_{z}(q_n(z)) = q_{n+1}(z) + q_{n-1}(z)$$ I want to solve this exactly and have been told it can be done with Bessel functions, but I do not see ...
0
votes
0
answers
53
views
Integral representation of $\text{Li}_0^{(1,0)}(z)$. Somos constant: $\sqrt{1\cdot\sqrt{2\cdot\sqrt{3\cdot...}}}$
I would like to represent Somos' quadratic recurrence constant with an integral.
$$\sigma_S=\sqrt{1\cdot\sqrt{2\cdot\sqrt{3\cdot...}}}=\prod_{k=1}^{\infty}\sqrt[2^k]{k}=\exp\left(\sum_{k=1}^{\infty}\...
- 2,373
0
votes
0
answers
18
views
Anger-Weber function for an integer value of the order
The Anger-Weber function is defined by
$$
A_{\nu}(z) = \int_0^\infty \exp\bigl(-\nu t - z \sinh(t)\bigr) \mathrm{d}t
$$
where $\nu, z \in \mathbb{C}$ with $\Re(z) > 0$.
I am not able to numerically ...
- 1,674
0
votes
0
answers
54
views
Numerical evaluation of the Schläfli integral
I'm trying to numerically evaluate
$$
S_{\nu}(z) = \int_0^\infty \exp\bigl(-\nu t - z \sinh(t)\bigr) \mathrm{d}t
$$
where $\nu, z \in \mathbb{C}$ with $\Re(z) > 0$. This integral is a part of the ...
- 1,674
0
votes
0
answers
31
views
Series expansion involving Kummer and Tricomi functions analogy
Good day to everyone. I've got in a pickle while toying around with some transformations.
It is well-known that the bivariate confluent hypergeometric function $\Phi_2(\cdot)$ can be expanded in the ...
- 51
1
vote
1
answer
239
views
+50
Addition formula for generalized Laguerre polynomials
For the Hermite polynomials, there is the following addition formula
Is there a similar formula for the generalized Laguerre Polynomials, in particular for $a=b=1/2$.
I.e. what is
$L^m_k(0.5x + 0.5 y)...
- 69
0
votes
0
answers
25
views
Cayley-Tranform is unitary equivalent to multiplication operator?
Let $H$ be a seperable komplex Hilbert space and
$A:D(A)\subset H \rightarrow H$ self adjoint then $C_A:=(A-i)(A+i)^{-1} \in L(H)$ is unitary and known as the Cayley transform of $A$.
Since the Calyey ...
- 622
1
vote
1
answer
115
views
Solve $\int_0^\infty\frac{\sin x}{x+1}dx$
I wonder how to solve $$\int_0^\infty\frac{\sin x}{x+1}dx$$I went about solving it like this: \begin{align*}\int_0^\infty\frac{\sin x}{x+1}dx& =\int_1^\infty\frac{\sin(x-1)}xdx=\int_1^\infty\frac{\...
- 4,490
0
votes
0
answers
80
views
Solve $y=\operatorname{erf}(x+c)+\operatorname{erf}(x-c)$ for $x$
$\newcommand{\erf}{\operatorname{erf}}\newcommand{\erfc}{\operatorname{erfc}}$
Is there a closed form solution for $y=\erf(x+c)+\erf(x-c)$?
More specifically, I want to solve
$$\erf(\frac{y}{\sqrt{2}})...
- 11.2k
-1
votes
0
answers
28
views
Finding the radius from a chord and arc, inverse sinc [duplicate]
This is based off some random YouTube comment. The person was asking about finding the radius of a circle when all you have is the length of an arc and the length of its chord. A respondent said it's ...
- 201
0
votes
0
answers
48
views
Coefficient of $x^n$ in Legendre series expansion
Suppose we are approximating a function $f$ with a Legendre series of order $N$, namely
$$
f(x) \approx \sum_{n=0}^N c_n P_n(x) \equiv f_N(x)
$$
where $P_n(x)$ is the $n^{th}$ Legendre polynomial and
$...
4
votes
2
answers
102
views
Equations similar to Lambert-$W$ with quadratic exponents
I've seen solutions saying that an equation in the format:$$ \ln(x) - \frac{bx}{a} = - \frac{bc}{a} $$
can be solved using the Lambert W function and I am comfortable doing so. My equation however is ...
11
votes
4
answers
448
views
Is it possible to integrate $\frac{ \tan ^{-1}(t)}{t^{2n}\,\sqrt{t^2-1}}$
Working on this question, I faced the problem of computing:
$$I_n=\int_1^a \frac{ \tan ^{-1}(t)}{t^{2n}\,\sqrt{t^2-1}}\,dt$$
For a given value of $n \geq 1$, Mathematica does not face any problem and ...
- 249k
2
votes
0
answers
100
views
Closed form for ${_4F_3}\!\left(\begin{array}c\tfrac12,\tfrac12,\tfrac12,\tfrac12\\\tfrac32,\tfrac32,\tfrac32\end{array}\middle|\tfrac12\right)$
I am trying to find the closed form expression for
$${_4F_3}\!\left(\begin{array}c\tfrac12,\tfrac12,\tfrac12,\tfrac12\\\tfrac32,\tfrac32,\tfrac32\end{array}\middle|\tfrac12\right).$$
I encountered ...
6
votes
1
answer
129
views
Evaluation of the integral of $\int_0^\infty \frac{\mathrm{Ti}_2(x)}{e^{2 \pi x} - 1}\,dx$
I am trying to evaluate the integral \begin{align*} \Omega \equiv \int_0^\infty \frac{\mathrm{Ti}_2(x)}{e^{2 \pi x} - 1}dx\end{align*} where $\mathrm{Ti}_2(x)$ denotes the inverse tangent integral. I ...
1
vote
0
answers
73
views
Analytic Solution for an Infinite Sum Involving 1F2 Hypergeometric Function with parameters whose difference is one half
I am attempting to find an analytic solution for the following infinite sum, where $x$ is a nonnegative real number:
\begin{equation}
\sum_{k=1}^{\infty} \frac{(-1)^k y^{2k} k}{(2k)!} \, {}_1F_2\left(...
- 11
5
votes
1
answer
158
views
Evaluating $\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{(ax)}\operatorname{arsinh}{(bx)}}{x}$ in terms of polylogarithms
Define the function $\mathcal{I}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ by the definite integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\right)}\...
- 29.2k
0
votes
1
answer
17
views
Gegenbauer functional equations
Let $C_n^\lambda$ be the Gegenbauer Polynomial with the parameter $\lambda$. We have the defining recurrence relation
$$nC_n^\lambda(x)=2(n+\lambda-1)x\,C_{n-1}^\lambda(x)-(n+2\lambda-2)C_{n-2}^\...
- 543
4
votes
1
answer
373
views
Integration of exponential of a function of cosines [closed]
I am trying to solve an integration in the form
$$\int_{0}^{2\pi} e^{a \cos{(\theta-b)} + c \cos{(2\theta)}} d\theta$$
where $a$, $b$, and $c$ are constants. I know that if $c=0$, the integration ...
- 75
9
votes
1
answer
218
views
Ratio of theta functions as roots of polynomials
I was playing with the theta functions with argument $ z = 0 $
$ \vartheta_2(q) =\sum_{n=-\infty}^\infty q^{(n+1/2)^2} $
$ \vartheta_3(q) =\sum_{n=-\infty}^\infty q^{n^2} $
$ \vartheta_4(q) =\sum_{n=-\...
- 571
6
votes
3
answers
210
views
Any other methods for evaluating $\int_{0}^{\frac{\pi}{4}}\frac{\arctan(\cos(x))}{\cos(x)}\,dx$
I evaluated the following integral:
$$I:=\int_{0}^{\frac{\pi}{4}}\frac{\arctan(\cos(x))}{\cos(x)}\,dx$$
I would like to see any alternate solutions, here is my work.
Using:
$$\frac{\arctan(x)}{x}\...
- 1,113
0
votes
2
answers
122
views
Find the answer to an series
I've been stuck with calculating the close form of series of the following problem.
\begin{align*}
\sum_{k=1}^{+\infty}\left(\ln(k)-\ln(x+k)+\frac{x}{k}\right)
\end{align*}
for real constant $x\geq1$...
- 405
4
votes
3
answers
267
views
On the integral $\int_0^1 \frac{\ln(2-x)}{1+x^2} \, \mathrm{d}x$
When I found closed form of integrals $1$ and $2$
\begin{align}
\int_0^1 \frac{\ln(1+x)}{1+x^2} \, \mathrm{d}x \tag{1}\\
\int_0^1 \frac{\ln(1-x)}{1+x^2} \, \mathrm{d}x \tag{2}\\
\end{align}
I ...
- 582
8
votes
0
answers
143
views
On $\zeta(5)$ and the closed form of $\sum_{n= 1}^\infty \frac{1}{n^5(e^{2\pi n}\pm1)}$?
Consider the closed-form of the sum,
$$\sum_{n= 1}^\infty \frac{1}{n^p(e^{2\pi n}\pm1)} = \; ??$$
for $\color{blue}{p=4m+1}$. (Since closed-forms are known for $p=4m+3.$) For $p=1$, we have,
\begin{...
- 52.3k
0
votes
0
answers
66
views
Is there a form for the inverse of $(1-l x) ^K(1+w x)^{(T-K)}$ in terms of "named" functions (Beta, Gamma, etc)?
Let $l \in (0,1)$, $w>0$, $K$ and $T$ positive integers.
The function $f(x) = (1-l x)^K(1+w x)^{(T-K)}$, restricted to $x \in [0,1]$ is then logconcave in x.
Thus, we can separate the domain into ...
- 1
22
votes
0
answers
298
views
What is so special about $\mathbb{Q}(\sqrt{398})$ that it has several good prime generating polynomials?
Going through an old post of mine from 2014, I realized there was a curiosity that hasn't been fully explained up to now. Consider the following simple prime-generating polynomials. (My thanks to ...
- 52.3k
1
vote
1
answer
517
views
On the vanishing of the integral $\int^{n+4}_0 dt P_{\ell}\left(1 - \frac{2t}{n+4}\right)(-t+2)_{n+1}$ for $\ell \geq n+2$
I came across the following integral
\begin{equation}
\int^{n+4}_0 dt P_{\ell}\left(1 - \frac{2t}{n+4}\right)(-t+2)_{n+1}
\, ,
\end{equation}
where in the above $P_{\ell}(x)$ is the Legendre ...
2
votes
2
answers
108
views
Does the second positive solution (besides $x = 1$) of the equation $e^{x^2-1}=x^3-x\ln x$ have a closed form?
Does the equation $$e^{x^2-1}=x^3-x\ln x$$ have a closed form solution ?
The given equation has $2$ positive real roots. Graphically
It is not hard to see that $x=1$ is a rational solution. The ...
- 486
0
votes
1
answer
77
views
Log-secant integral using exponential generating function $\int_0^{\pi/2}\log^n(\sec^2 x)\ dx$
I am trying to obtain a recurrence relation for the following integral,
$$I_n=\int_0^{\pi/2}\log^n(\sec^2 x)\ dx$$
using exponential generating functions.
The general structure of the integral seems ...
- 2,020
11
votes
1
answer
619
views
Asymptotics of $\displaystyle{\sum_{i=0}^n\sqrt{i(n-i)}}$ as $n\to\infty$
I am studying a bit of asymptotics and for practice I decided to find the asymptotic of the following,
$$s(n)=\sum_{i=0}^n\sqrt{i(n-i)}$$
as $n\to\infty$. This comes directly from this post, where @...
- 2,020
3
votes
1
answer
157
views
Closed form for $\sum_{n=1}^\infty\binom{2n}{n}\zeta(m,n+1)x^n$
I need a closed form for the following series which appeared when evaluating another sum.
$$S(m,x)=\sum_{n=1}^\infty\binom{2n}{n}\zeta(m,n+1)x^n, \quad m\in\mathbb{Z}$$
I've tried expanding the ...
- 2,020
2
votes
0
answers
58
views
Evaluating $\int\frac{\log(x+a)}{x}\,dx$ in terms of dilogarithms
As per the title, I evaluated
$$\int\frac{\log(x+a)}{x}\,dx$$
And wanted to make sure my solution is correct, and if not, where I went wrong in my process. Here is my work.
$$\int\frac{\log(x+a)}{x}\,...
- 1,113
0
votes
1
answer
61
views
Series expansion of the power Reciprocal gamma function
Based on the post Non-recursive closed-form of the coefficients of Taylor series of the reciprocal gamma function, is it possible to obtain some Taylor expansion series for
$$
f(z)=\frac{1}{[\Gamma(z)]...
- 179
7
votes
2
answers
232
views
About the Integral $\int\arcsin\left(\sin^{2}x\right)dx$
$$\int\arcsin\left(\sin^{2}x\right)dx$$
I am not able to find a closed form elementary solution for this, though I have no reason to believe it exists.
But trying out the Definite Integral as follows:...
- 1,723
1
vote
1
answer
58
views
Series expansion for $\displaystyle{\psi(t+2)-\psi\left(\frac{t+3}{2}\right)}$
I need the power series (starting with index $k=1$) of a difference between two digamma functions.
$$\psi(t+2)-\psi\left(\frac{t+3}{2}\right)=\sum_{k=1}^\infty a_kt^k$$
In otherwords I want $a_k$ in ...
- 2,020
2
votes
0
answers
28
views
How to identify a Fox $H$ function from its Mellin transform?
I have obtained a Mellin transform
$
\mathcal{M}[f](s)= \frac{ \Gamma\left( 1-\frac{2-s}{a}\right) \Gamma\left( \frac{2-s}{a}\right) \Gamma\left( \frac{s}{2}\right) {{2}^{s-1}}}{a \Gamma\left( 1-\...
- 1,134
3
votes
0
answers
148
views
How to finish the integral $\int_{0}^{\infty}\frac{\sin(x)}{\sinh(x)}\,dx$ [duplicate]
I was evaluating:
$$I:=\int_{0}^{\infty}\frac{\sin(x)}{\sinh(x)}\,dx$$
This is what I did, how can my answer be simplified if correct. The following is my work:
$$I=2\int_{0}^{\infty}\frac{\sin(x)}{e^...
- 1,113
0
votes
0
answers
60
views
Analytic Form of $\int dx \frac{e^{ - m\sqrt{x^2+b^2}}}{\sqrt{x^2 + b^2}}e^{ikx }$
Does anyone know how to get an analytic expression of this?
$$\int^L_{-L} dx \frac{e^{- m\sqrt{x^2+b^2}}}{\sqrt{x^2 + b^2}}e^{ikx} $$ where $k = \dfrac{n\pi}{L}$, $m$, and $b$ are real parameters.
or $...
4
votes
2
answers
266
views
Integral over a product of polynomial, exponential and Bessel function
In a physics textbook I'm working through I found an interesting integral identity which I want to prove:
\begin{equation}
\int_0^\infty t^{\nu +1} J_\nu(\beta t) e^{-\alpha t} \, dt = \frac{2\alpha (...
- 343
2
votes
0
answers
36
views
Explicit example of an entire function with simple zeros at precisely the square roots of the positive half integers
I'm looking for an entire function with the property that $f(\sqrt{n+1/2}) = 0$ for $n=0,1,2,\dots$, all of which are simple zeros and $f$ has no other zeros.
I know that such functions exist and can ...
- 93
1
vote
1
answer
102
views
D-dimensional Fourier transform of $\exp{(-c|\vec{x}|^\alpha})$
Prove that for $\vec{x},\vec{\xi} \in \mathbb{R}^D$, $\mathbb{R} \ni c, \alpha = \mathrm{const.}$ the $D$-dimensional Fourier transform of $ f(\vec{x}) = \exp{(-c|\vec{x}|^\alpha})$
\begin{equation}
\...
- 343
7
votes
6
answers
701
views
A question on Beta function
I need an asymptotic expansion/closed form for $$\sum_{k=1}^{\infty}\int_{0}^{\infty}(B(x+n+k,n+1))^2\ dx$$ where $B(m,n)$ is the Beta function and $n\in\mathbb{N}$.
Denote $$I_n=\sum_{k=1}^{\infty}\...
- 304
0
votes
0
answers
26
views
What are some formulas involving Dottie number and digamma function?
Here are two formulas that involve Dottie number and Gamma function:
$\Gamma \left( \frac{1}{2} + \frac{d}{\pi} \right) \Gamma \left( \frac{1}{2} - \frac{d}{\pi} \right) = \frac{\pi}{d}
$
and
$\sqrt{\...
- 8,695
1
vote
0
answers
55
views
Questions on Gauss's geometric interpretation of spherical functions
(This question was initially posted on HSM stackexchange, after that I came to conclusion that it is too mathematical to be answered there and asked it on mathoverflow. However, it recieved no ...
- 974