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

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0
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0answers
19 views

Simplify $\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,J_1(kR)\frac{1}{1+bk^2}$

EDIT: I would love to find an analytical solution for this definite integral: $$\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,J_1(kR)\frac{1}{1+bk^2}$$ with $\delta>0,\, R>0,\,b>0$. Does ...
0
votes
0answers
16 views

Infinite product representation for the Sine Integral $\mathrm{Si}(z)$

The infinite series representation of the sine integral (http://en.wikipedia.org/wiki/Trigonometric_integral, previous m.se question: Is there any infinite series representation of the sine ...
0
votes
0answers
7 views

Legendre's Chi- Function

I want to get the numerical value(twenty at thirty decimals) of $$\operatorname\chi_{2}(\frac{1}{\sqrt{2}})$$ Thanks you very much.
0
votes
0answers
6 views

What is the region ( area) of integration in Double mellin Barnes integral?

What is the region ( area) of integration in Double mellin Barnes integral ? In H-function of two variables we are using double Mellin-Barnes contour integration on s and t planes where s and t are ...
4
votes
1answer
57 views

Infinite sum of products of four Bessel functions

The discrete Schrödinger equation for two interacting electrons in 1D under an electric field reads $$ E\psi_{mn}=[(m+n)F+U\delta_{mn}]\psi_{mn}-\psi_{m+1,n}-\psi_{m-1,n} -\psi_{m,n+1}-\psi_{m,n-1}\ . ...
0
votes
1answer
14 views

Generalizing 1D function to higher dimensions

I have a function in 1D given by $f(x) = \tanh(x-x_1) + \tanh(x-x_2)$. I want to generalize this to two dimensions, such that it describes a circle. The function $f(x,y)$ has to have a form such that ...
1
vote
1answer
43 views

get a integral from another

if $\int\limits_{0}^{+\infty}x^3e^{-\alpha x^2} dx=\frac{1}{2A}$ then $\int\limits_{0}^{+\infty}x^4e^{-\alpha x^2} dx=$ i tried to use integration by parts $$\begin{align} ...
2
votes
0answers
36 views

Comprehensive summary of where the function $\pi^{-\frac x\pi}$ can be encountered

I am studying the special functions, including the Riemann Xi and Zeta, and everywhere a function $\pi^{-\frac x\pi}$ pops up, usually as multiplier to the Gamma function. But yet I am not sure this ...
2
votes
0answers
31 views

Definite integral of a hypergeometric function of an imaginary argument

How would one deal with such an integral? $$\int_0^\infty\frac{e^{-n r}}{r}{}_1F_1(i/k+1;2;2i kr) \, \mathrm{d} r$$ Here $F$ is the confluent hypergeometric function, $n\in\mathbb{N}$ and $k>0$ ...
0
votes
0answers
27 views

evaluation of the limit $\lim_{u\to 0}\phantom{ }_0F_1(1,-u)u^{1-\alpha/2}$

What are the values of the limits \begin{equation} \lim_{u\to 0}\phantom{}_0F_1(1,-u)u^{1-\alpha/2}=?\\ \lim_{u\to \infty}\phantom{}_0F_1(1,-u)u^{1-\alpha/2}=? \end{equation} where ...
1
vote
1answer
22 views

Where can I find simple integration problems (and other computational exercises) involving special functions?

Working lots of computational exercises in my pre-calculus and calculus classes has given me a great deal of intuition in dealing with elementary functions. Thanks to these years of practice, I can ...
2
votes
0answers
31 views

$\frac{\sin(nx)}{\sin(x)}=(-4)^{(n-1)/2} \prod_{1\leq j \leq (n-1)/2}(\, \sin^2(x)-\sin^2(\frac{2\pi j}{n})\,)$

In Serre's A Course in Arithmetic, it states For $n$ odd and positive integer, proof that $\frac{\sin(nx)}{\sin(x)}=(-4)^{(n-1)/2} \prod_{1\leq j \leq (n-1)/2}(\,\sin^2(x)-\sin^2(\frac{2\pi ...
-1
votes
1answer
49 views

Write$\frac{\sin(nx)}{\sin(x)}$ as polynomial in $\sin^2(x)$ [closed]

How to write $\frac{\sin (nx)}{\sin(x)}$ as a polynomial of degree $\frac{(n-1)}{2}$ in $\sin^2(x)$, where $n$ is a positive odd number.
2
votes
1answer
30 views

Limit of Ratio of Chebyshev Polynomials

I have been trying to compute the limit $$\lim_{n\to\infty}{{U_n(x)^2}\over{U_{n-1}(x)^2+U_n(x)^2}}$$ where $U_n(x)$ is the $n$-th Chebyshev polynomial of the second kind and $x\ge 1$. Using software ...
2
votes
0answers
16 views

How to find a $\theta$ function verifying this property?

Let $r>4$ and $n>1$ be positive integers. Intuitively, the infinite sum $$S=\sum_{m=1}^{∞}\frac{2m}{r^{m^2}}$$ is related to a $\theta$ function. However, I cannot find a way to calculate this ...
2
votes
1answer
74 views

finding a harmonic sum using residues/complex analysis

Evaluate: $$S = \sum_{n=1}^{\infty} \frac{H_n}{n^2}$$ Using complex analysis. I just needs hints, I have no attempts, but I believe is has to do with residues.
0
votes
0answers
8 views

Integrals involving Marqum Q and Gaussian functions

I want to evaluate the following double integral. Any closed form solution exists? $$I(x)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-y^2}e^{-z^2} Q_1 \left[\sqrt{y^2+z^2},\sqrt{2x}\right]^2 ...
3
votes
1answer
54 views

the integral of $\sin(z^2) \exp\left({-4z^2xy \over y^2-x^2}\right)$ can be written in Fresnel integrals?

$$\int_{0}^{(y^2-x^2)/ 4t}s^{-1/2} \sin(s) \exp\left({-4sxy \over y^2-x^2}\right)\mathrm{d}s=2\int_{0}^{\sqrt{(y^2-x^2)/4t}}\sin(z^2) \exp\left({-4z^2xy \over y^2-x^2}\right)\mathrm{d}z$$ I applied ...
3
votes
0answers
23 views

Software to compute spherical harmonics in higher than 3 dimensions (100 or maybe 500 dimensions)?

I have been trying to find an implementation of Spherical harmonics for higher dimensional data but I couldnt find anything in Sage, Mathematica, Matlab. Does anyone have any idea of a standard/fast ...
3
votes
1answer
25 views

How to Derive this Digamma identity?

I dont see the transition from $(-z)^k$ in the fist sum to the transition to $(z+2)^k$ in the second sum? How is that derived?
2
votes
0answers
22 views

Residue of $f(z)$ using Laurent Series at $z=-2$ [duplicate]

Calculate the residue of: $$f(z) = \frac{\psi(-z)}{(z+1)(z+2)^3} \space \text{at} \space z=-2$$ Where $\psi(z)$ is the digamma function, and $\zeta(z)$ is the Riemann-zeta function (below). The ...
0
votes
0answers
38 views

Lambert W function, W(x), representation for entire domain

The Taylor series for the Lambert W function is $W_0(x)=\sum_{n=1}^\infty\frac{(-n)^{n-1}x^n}{n!},\left\{x\in\mathbb{R}:|x|<\frac{1}{e}\right\}$. Is there any exact closed form way to express ...
1
vote
2answers
40 views

Derivated function $f$ so that $f(x+y)=f(x)f(y)$ and $f'(x)f(y)=f(x)f'(y)$ for all $x,y \in \Bbb R$

Let $f:\Bbb R\to \Bbb R$ a derivated function in all $\Bbb R$ that satisfies the condition $$f(x+y)=f(x)f(y),\;\,\,\text{for all $x,y \in \Bbb R$}$$ I already tried that $f'(x)f(y)=f(x)f'(y)$ for ...
0
votes
1answer
17 views

Question Mark Function and continued fraction representations

One could defined Minkowki's question mark question by : $$?(x) = a_0 + 2 \sum_{n= 1}^\infty \dfrac{(-1)^{n+1}}{2^{a_0 +\dots +a_k}},$$ with $x = [a_0;a_1,a_2,\dots]$. Is Minkowski's question mark ...
3
votes
0answers
87 views

Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$

Prove that: $$ I=\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx=\frac{7}{2}\zeta(3){\log^22}-\frac{\pi^2}{6}{\log^32}-\frac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\frac{\pi^4}{48}\ln2 $$ Using ...
2
votes
0answers
33 views

Intuitive explanation why in some contexts logarithm shifted by Euler-Mascheroni constant is more natural

Natural logarithm is defined as inverse function to exponent. This way defined it has the value of $0$ in $x=1$. But if we define natural integral the following way ...
2
votes
2answers
107 views

From $P(x;W) = \frac{1}{Z(W)} \exp \bigl[ \frac{1}{2} x^T W x \bigr]$ to Sigmoid

In a book chapter that talks about the Boltzmann distribution, $$ P(x;W) = \frac{1}{Z(W)} \exp \bigg[ \frac{1}{2} x^T W x \bigg] $$ where $W$ is symmetric with zero diagonal. It makes a seque ...
0
votes
1answer
19 views

Solution of Bessels differential equation

What is the solution of the of the differential equation $x^{2}y''+xy'+\left(4x^{2}-\dfrac{9}{25}\right)y=0$ in terms of Bessel's polynomial of the form $y=AJ_{n}(x)+BJ_{-n}(x)$, where $A$ and $B$ are ...
4
votes
2answers
94 views

Approximation for elliptic integral of second kind

My (physics) book gives the following approximation: $\int_{-\pi/2}^{\pi/2} \sqrt{1-(1-a^2) \sin(k)^2} dk \approx 2 + (a_1 - b_1 \ln a^2) a^2 + O(a^2 \ln a^2)$ where a1 and b1 are "(unspecified) ...
6
votes
2answers
135 views

Computing a nasty integral (probably with computer algebra system)

I'm trying to do this integral, not sure if it is possible: $$ \int_{1}^{\infty}\int_{0}^{\infty} \exp\left(\, -\,{x^{2} \over y^{2}}\,\right) \exp\left(\,-\,{y^{2} \over z^{2}}\,\right) \exp\left(\, ...
1
vote
2answers
116 views

Intersection of functions $\ln(x)$ and $\frac{1}{x}$

How to find $x$ such that $$\ln(x)=\frac{1}{x}$$ Thank you!
2
votes
1answer
43 views

Is it possible to express $\Gamma\!\left(\tfrac{1}{50}\right)$ through values of the $\Gamma$-function at rational points with smaller denominators?

Sometimes it is possible to express a value of the $\Gamma$-function at a rational point through values of the $\Gamma$-function at rational points with smaller denominators, e.g. ...
-3
votes
0answers
64 views

Help to resolve this equation?

How do I solve the following equation for $x$ ...
1
vote
1answer
51 views

Limit of positive sum is negative? Related to polylgarithm

So my initial point of confusion is on \begin{equation} \lim_{x\rightarrow\infty} \ x+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\cdots \end{equation} which we recognise as \begin{equation} ...
1
vote
2answers
48 views

Changing a sigmoid curve to have an adjustable point of inflection

I am trying to an implement an adjustable Sigmoid curve such as in the YouTube video here. I found a potentially good candidate: $$f_k(x) = \frac{\left(x-x\cdot k\right)}{k-\left|x\right|\cdot 2\cdot ...
1
vote
1answer
96 views

Is $ d^m_xP_l(x) d^{m+1}_xP_{l+1}(x)- d^m_xP_{l+1}(x) d^{m+1}_xP_{l}(x)$ positive?

Does the expression $$ d^m_xP_l(x) d^{m+1}_xP_{l+1}(x)- d^m_xP_{l+1}(x) d^{m+1}_xP_{l}(x)$$ always have a fixed sign ( so is it always positive or negative) on the interval(-1,1)?. $P_l$ is the l-th ...
0
votes
0answers
35 views

Can this series be expressed as a Hyper Geometric function

I am trying find a Hyper Geometric function representation of the following series. $$\sum\limits_{k=0}^{\infty} \frac{a^k}{k!}\frac{\Gamma \left(\frac{b+k}{c}\right)}{\Gamma ...
1
vote
0answers
52 views

Closed form for generating function of Riemann Xi function

What is the closed form for $$f(x)=\ \sum_{k=1}^\infty \frac{\xi(k)x^k}{k!}$$ or $$g(x)=\frac12 \sum_{k=1}^\infty \frac{\xi(k+1/2)x^k}{k!}$$ or $$w(x)=\frac12 \sum_{k=1}^\infty ...
0
votes
0answers
23 views

Uniform limits of pathological functions

I'm interested in the following (perhaps somewhat artificial) problem: Suppose $f$: $\mathbb{R}\longrightarrow\mathbb{R}$ is a function taking open subsets $U\neq\emptyset$ to dense subsets. It's ...
0
votes
1answer
38 views

Simplify Infinite Series Involving Gamma Function $\Gamma$

This question originated from this problem. Can anyone help me simplifying the infinite series below: $$\sum_{n=0}^{+\infty}\frac{1}{\Gamma(\beta_2-n)}\frac{(-e^{-t})^n}{n!}$$ The only idea I have ...
6
votes
2answers
99 views

Evaluating $\int_0^\infty \sqrt{\frac{x}{e^x-1}}dx$ in terms of special functions

Introduction: I've been studying integrals of the form $$\int_0^\infty \frac{x^a}{(e^x-1)^b}dx$$ where a and b are real parameters. I've been able to find closed forms for the integral in terms of the ...
3
votes
1answer
79 views

Prove that $\int_0^{\infty} \int_0^{\infty} e^{-(x^2+y^2+2xy \cos \theta)} \,dx dy = \frac{\theta}{2\sin\theta}$

Prove that the following integral: $$\int_0^{\infty} \int_0^{\infty} e^{-(x^2+y^2+2xy \cos \theta)} \,dx dy = \frac{\theta}{2\sin\theta}$$ The hints written on the book are beta function and to ...
2
votes
2answers
39 views

A Generating function of product of binomial coefficients

Are any of you familiar with the closed form solutions for $$\sum_{j=0}^\infty \binom{j+\alpha+\beta}{j+\alpha} \binom{\beta}{j}x^j $$ where $\alpha$ and $\beta $ are integers? Thanks!
2
votes
2answers
127 views

Solving the equation $\ln(x)=-x$

I tried solving this equation for a long time but did not succeed. Any help is appreciated. $$\ln x=-x$$ I am not sure the tag is correct, I am not familiar with English mathematical terms. Please ...
0
votes
0answers
47 views

Reducing integral

let $$I=\int \frac{dx}{\sqrt{mx^3-x^2+n}}$$ How do we reduce $I$ to an elliptic Integral of the first Kind ? where $m,n>0$ are constants.
1
vote
0answers
33 views

How to compute $\int_{-1}^1 x^p (1-x^2)^{\frac{d-3}{2}} P_n^d(x) dx$

For a project I want to get a closed form solution of $$\int_{-1}^1 x^p (1-x^2)^{\frac{d-3}{2}} P_n^d(x) dx$$ Here $p \in \mathbb{N},\; d\ge3, \; d\in\mathbb{N}$ and $P_n^d$ is the associated ...
0
votes
0answers
21 views

Associated Legendre polynomial expansion of $\exp(\xi)$

For a project I need to compute the coefficients of the Associated Legendre polynomial expansion of the $\exp$ function. That is I need to find $b_n$ such that $$\exp(x^Ty) = \exp(\xi) = ...
4
votes
1answer
42 views

How I can simplify this inequality or how I can solve it?

How I can simplify this inequality or how I can solve it: $$\left\lceil\dfrac{\ln(t+2)}{\ln 2}\right\rceil-\left\lfloor\dfrac{\ln(t+1)}{\ln2}\right\rfloor>1$$ where $t$ is a positive integer. ...
1
vote
0answers
13 views

Integral involving the Legendre Polynomials

I'm trying to compute the following integral: $\int_a^{+\infty} \frac{dt}{(P_\lambda(\tanh{t}))^2}$ and to be honest I have no idea on how I should attack this problem. Is there any reference that I ...
0
votes
1answer
27 views

Is $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} {(\delta+1)^n}$ for any $n$ ?(in this specific case)

Let $\alpha=(K-1)a$, $\beta=K$ and $\delta=Ka$, where $K>a\ge 1$ ($\delta>\alpha>\beta$). Can we claim that $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} ...