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

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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
88 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 ...
6
votes
2answers
136 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(\, ...
3
votes
1answer
324 views

Incomplete Fermi-Dirac integrals and polylogs

The complete Fermi-Dirac integrals $$ F_s(x) = \frac{1}{\Gamma(s+1)} \int\limits_{0}^{\infty} \frac{t^s}{e^{t-x}+1} \: dt $$ are related to the polylogarithms, see http://dlmf.nist.gov/25.12#iii $$ ...
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
3answers
170 views

Integral $\int_0^1 \frac{x\log x+1-x}{x \log^2 x}\log(1+x)\, dx=\log\frac{4}{\pi}$

Hi I am trying to prove this $$ I:=\int_{0}^{1} {x\log\left(\,x\,\right) + 1 - x \over x\log^{2}\left(\,x\,\right)}\, \log\left(\,1 + x\,\right)\,{\rm d}x=\log\left(\,4 \over \pi\,\right). $$ Thanks. ...
4
votes
1answer
216 views

Evaluating $\int \arccos\bigl(\frac{\cos(x)}{r}\bigr) \, \mathrm{d}x$

The title says it all, really - I am looking for $$\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$$ where $0<r<1$ and $x$ is in a domain where the integrand is real. It came up ...
2
votes
1answer
158 views

Special functions as representations of Lie Groups

-The spherical harmonics $Y_{lm}$ are complete on $L^2(S^2)$. They are also a representation of the (compact) Lie group $SO_3 (\mathbf{R})$. -The functions $e^{i n x}$ are complete on ...
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!
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 ...
1
vote
2answers
49 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 ...
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. ...
2
votes
1answer
266 views

Solve equation with lower gamma function: $A \gamma(2;x/B)=x$ for $x$

I need to find an expression for $x$ given: $A \gamma(2;x/B)=x$ where $\gamma(a,x)=\int\limits_0^x t^{a-1} e^{-t} \mathrm{d}t$ is the lower incomplete gamma function. $A$ and $B$ are real, positive ...
-3
votes
0answers
64 views
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} ...
3
votes
3answers
855 views

Continuous function with local maxima everywhere but no global maxima

Can there be such a function: $f \colon \mathbb R \to \mathbb R$ is continuous and non-constant. It has a local maxima everywhere, i.e., for all $x \in \mathbb R$ there is some $\delta_x>0$ such ...
0
votes
2answers
125 views

Intepolate from linear to step function, and one application for shading colors

I'm running after a particular function $f_\sigma : [−1,+1] \rightarrow [-1,+1]$ that could take three different forms depending on the value of its parameter $\sigma$. Could anyone help me ...
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 ...
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 ...
0
votes
1answer
39 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 ...
9
votes
3answers
610 views

Limit of Zeta function

I'm looking for a reference for (or an elementary proof of) $$ \lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$ Thanks for your help.
3
votes
1answer
307 views

Integrals $\int\limits_0^t {{e^u}\log u\operatorname d \!u} $ and $\int\limits_0^t {{e^{ - u}}\log u\operatorname d \!u} $

Ok, I want to find $$\int\limits_0^t {{e^u}\log udu} $$ and $$\int\limits_0^t {{e^{ - u}}\log udu} $$ I'm thinking as follows $$d\left( {{e^u}\log u} \right) = {e^u}\log udu + ...
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
1answer
270 views

Analytic continuation of factorial function

We know that the factorial can be extended to the whole complex plane except at negative integers and $0$ . But are there any theorems that allow us to do so ? . I know we can use the Identity ...
2
votes
1answer
39 views

Large-z limit of the *other* second derivative of the Laguerre polynomial

I'm trying to find the asymptotic behavior of the second derivative of the Laguerre polynomial (more precisely, the associated analytic function), $\frac{\partial}{\partial n^2}L_{n}(z)$, as $z\to ...
2
votes
2answers
129 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 ...
40
votes
1answer
1k views

Extending the result $\int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi} $

While generalizing this result, I succeeded in proving that for $\alpha > 0$, $\beta < 1$ and $1 < 2\alpha + \beta < 3$, we have \begin{align*} &\int_{0}^{\infty} \left[ \left( ...
17
votes
4answers
824 views

Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$

Background: Evaluation of $\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$ We can prove using the Beta-Function identity that $$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma ...
2
votes
2answers
112 views

A infinite sum with harmonic serie

Proof or disproof the folowing statement: $$\sum_{n=1}^{+\infty}\frac{2n+1}{(n^2+n)^2}H_n=\sum_{n=1}^{+\infty}\frac{1}{n^3}$$ where $\displaystyle H_n=\sum_{k=1}^{n}\frac{1}{k}$.
6
votes
2answers
164 views

Closed form for a zeta series

It is not that diffcult to derive \begin{align} \sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{k2^k}=&-\frac{\gamma}{2}+\ln\left(\frac{2}{\sqrt{\pi}}\right)\tag{1}\\ ...
9
votes
1answer
142 views

Closed-form of sums from Fourier series of $\sqrt{1-k^2 \sin^2 x}$

Consider the even $\pi$-periodic function $f(x,k)=\sqrt{1-k^2 \sin^2 x}$ with Fourier cosine series $$f(x,k)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n \cos2nx,\quad a_n=\frac{2}{\pi}\int_0^{\pi} ...
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.
8
votes
2answers
228 views

What is the closed form of $\sum _{n=1}^{\infty }{\frac { {{\it J}_{0}\left(n\right)} ^2}{{n}^4}}$?

Using Maple I am obtaining the numerical approximation $$0.5902373619$$ Please, let me know what is the closed form. Many thanks.
19
votes
0answers
118 views

Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
8
votes
2answers
128 views

Closed form for integral of integer powers of Sinc function

(Edit: Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.) For all $n\in\mathbb{N}^+$, ...
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
44 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 ...
7
votes
3answers
310 views

An infinite product for $\left(\frac{\eta(13\tau)}{\eta(\tau)}\right)^2$?

Given the Dedekind eta function, $$\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$$ where $q = \exp(2\pi i\tau)$. Consider the following "family", $\begin{align} ...
0
votes
0answers
27 views

Analyze the variation of $f(x)=(1+\frac{1}{x})^{-K}$ $_2F_1((K-1)a,K,Ka,\frac{1}{1+x})$ w.r.t. $x$

Is there any way to analyze the variation(w.r.t. $x$) of the following function: $f(x)=(1+\frac{1}{x})^{-K}$$ _2F_1((K-1)a,K,Ka,\frac{1}{1+x})$, where $ _2F_1$ is the Gauss' Hypergeometric Function, ...
1
vote
1answer
80 views

$\lim_{s\to 0^+}\int_0^\infty a(t) e^{-st} dt $

$$\int_0^\infty a(t) e^{-st} dt = f(s)$$ What is the meaning of the limit of this integral as $s\to 0^+.$
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} ...
0
votes
0answers
58 views

Definite integral involving Error function

Let us write $$\mathrm{erf}(x)=\frac{2}{\sqrt {\pi}}\int_0^x e^{-t^2}dt $$ for the usual Gauss error function. Given natural numbers $m,n,k$ I am interested in computing the integral ...
9
votes
2answers
279 views

Integral $\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}$

$$ I:=\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}. $$ Thank you. The Gamma function is given by $\Gamma(n)=(n-1)!$ and its integral representation is $$ \Gamma(x)=\int_0^\infty t^{x-1} ...
0
votes
1answer
25 views

On the continuity of $xf(x)$ and $x^2f(x)$, where $f$ is the Dirichlet function

Let $$f(x) = \begin{cases}1\qquad x\in\mathbb{Q}\\ 0\qquad x\notin\mathbb{Q} \end{cases}$$ Then how do I show that $xf(x)$ is continuous in $0$ and that $x^2f(x)$ is differentiable there as well? ...
5
votes
1answer
71 views

Sum involving zeros of Bessel function

I came across the following sum in my work involving the infinite sum of function of zeros of Bessel functions. $$ \displaystyle ...