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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2answers
41 views

Why is the imaginary part of the logarithm of the gamma function a square wave?

I just stumpled upon it and it made me curious. Why is the imaginary part of $\ln(\Gamma(x))$ a square wave for $x < 0$ ? The square wave has a period of 2 and a amplitude of $\pi/2$. How can one ...
4
votes
2answers
41 views

Simplification of a combination of 6 values of the gamma function

I'm trying to simplify this combination of gamma functions: ...
1
vote
1answer
35 views

$\zeta_m(s)=\prod\limits_{p\nmid m} \frac{1}{\left(1-\frac{1}{p^{f(p)s}}\right)^{g(p)}}$ is a Dirichlet series with non-negative coefficients

Let $p$ be a prime number, $m$ be any integer, $f(p)$ be the order of $p$ in $(Z/mZ)^*$, $i.e.$ $p^{f(p)} \equiv 1 \pmod m$ with $f(p)$ smallest. Let $g(p)=\frac{\phi(m)}{f(p)}$ is a integer where ...
6
votes
0answers
56 views

Function preserving exponentiation

I'm wondering what kind of function preserves exponentiation, i.e., what is an $f$ such that $f(a^b)=f(a)^{f(b)}$?
0
votes
0answers
19 views

Function property of $o$

I have a doubt regarding $o$-function. Could we write $o(\|\theta h)\|)=\theta \ o(\|h\|)$ ?
1
vote
2answers
38 views

Solution to simple functional equations

What is $\psi$ in functional equation: $$\sum _{n=0}^{\infty }{\frac { \left( -1 \right) ^{n}}{n+x}}=1/2\,\Psi \left( 1/2+x/2 \right) -1/2\,\Psi \left( x/2 \right)?$$
1
vote
0answers
15 views

Important Functions That Are Multivariable Integrals

There are lots of "important" functions of one variable that are defined in terms of integrals for which no closed form exists, like the Gamma Function and the normal distribution. Are there any such ...
0
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1answer
69 views

Definite integral involving modified bessel function of the first kind and its logarithm

I'm trying to solve the following integral $$ T=\int_0^\infty \exp(-a x^2) I_1(b x) \log(I_1(b x))\, dx $$ where $I_1(x)$ is the modified Bessel function of the first kind and order one, and $a$, $b$ ...
0
votes
0answers
30 views

on convergence of Integral involving Bessel function

For $r>0$ suppose $$\chi(r)=\sqrt{r}J_{\nu}(re^{i\frac{\pi}{4}})$$ where $J_{\nu}$ is a Bessel function of order $\nu$ (which is complex number). I am trying to find for what values of $\nu$ is ...
3
votes
3answers
117 views

Which methods can be used to evaluate the following integral?

How can I evaluate the following integral $$ \int_{0}^{\infty} x^{-1/2} \exp({-x/2})\ dx $$ I know the answer is $\sqrt{2\pi}$.
4
votes
2answers
79 views

Simplify $\frac{\Gamma(n)}{\Gamma(n+a)}$ with $a\in\mathbb C$.

How can simplify the following expression? $$\frac{\Gamma(n)}{\Gamma(n+a)}\sim \cdots\text{ ?}$$ Where $a\in\mathbb C$, $n\in \mathbb N$. Any suggestions please? I propose the following. We have ...
1
vote
1answer
56 views

Does a function relationship for a specific $y$ hold for any?

Let $f:\mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$ , continous $\forall$ $y$ in $\mathbb{R}$, $\exists$ $a,b$ such that $f(x,y) = ax + b$ $\forall$ $x$ in $\mathbb{R}$, $\exists$ $a',b'$ such ...
2
votes
0answers
51 views

Fourier transform all steps walkthrough for wave vector $k$ and $x$

Below is my walkthrough of a fourier transform. My problem is that I want to do all the similar steps for a fourier transform between position x and the wave vector k. That is working on a solution of ...
1
vote
1answer
33 views

is split function derivable

$ f(x) = \begin{cases} \frac{sin(x)}{x}, & x \ne0 \\ x+1, & x=0 \end{cases}$ I know that the function is a continuous function in R. But is this function derivable at x=0? I am not sure.. ...
1
vote
0answers
52 views

Rodrigues formula Associated Laguerre polynomial

Could you find the rodriguez formula of $$L_n^{\beta }\left(x^2\right)$$ knowing that $$\frac{\left(e^x x^{-\beta }\right) \frac{\partial ^n\left(e^{-x} x^{\beta }\right)}{\partial ...
7
votes
2answers
195 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
123 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
36 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 ...
1
vote
0answers
33 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
1answer
41 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
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0answers
20 views

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

In double Mellin-Barnes contour integrals how to decide the region of integration?
4
votes
1answer
100 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
19 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
48 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
43 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
1answer
179 views

Definite integral of polynomial times exponential times hypergeometric function of imaginary argument

How would one deal with such an integral? $$I(k)\equiv \int_0^\infty r^n e^{-r(1+\mu)} e^{-{\mathrm i} kr}\:{}_1F_1({\mathrm i}/k+1;2;2{\mathrm i} kr) \, \mathrm{d} r$$ Here $n\in\{0,1\}$, $\mu\in ...
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
31 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
1answer
52 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 ...
2
votes
1answer
57 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
19 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
90 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
25 views

Integrals involving Marcum Q and Gaussian functions

I want to evaluate the following double integral. Does any closed form solution exist? $$I(x)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-y^2}e^{-z^2} Q_1 ...
3
votes
1answer
69 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
36 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
32 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 ...
1
vote
0answers
65 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
44 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
22 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 ...
5
votes
0answers
117 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
37 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
114 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
26 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
130 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
147 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
126 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
62 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. ...
1
vote
1answer
58 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
122 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 ...