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

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3
votes
0answers
23 views

how to compute this limit

compute $I=\lim\limits_{n\to+\infty}\sqrt[n]{\int\limits_0^1x^{n+1}(1-x)\cdots(1-x^n)dx}$ attempt: i tried to evaluate the integral $$\begin{align} ...
2
votes
0answers
15 views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
0
votes
0answers
24 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
17 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
12 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
63 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
16 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
44 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
26 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
32 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
56 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
35 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
17 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
118 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
128 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
48 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. ...