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

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12
votes
1answer
234 views

The log integrals $\int_{0}^{1/2} \frac{\log(1+2x) \log(x)}{1+x} \, dx $ and $ \int_{0}^{1/2} \frac{\log(1+2x) \log(1-x)}{1+x} \, dx$

Using M.N.C.E.'s suggestion in the comments, both integrals can be expressed in terms of integrals that can be evaluated by integrating by parts. In attempting to evaluate $ \displaystyle ...
1
vote
1answer
27 views

Can anyone come up with an example of a monotonously not-falling function whose breakpoints are everywhere dense set on $[0,1]$?

$$f: \displaystyle{R\to[0,1]}$$ I can't come up with one, can anyone else ? :D How about : $$f(x)=\begin{cases} x, x \in Q \\ 0, x\in I \end {cases}$$ ??
1
vote
0answers
130 views

How to compute the definite integrals of special functions?

How can these integrals be solved: $${1\over \pi} \int_{0}^{\infty}\left({{\sqrt{x}(a-bx)}\over {x^{3}+(a-bx)^{2}}}\right)\cos(\sqrt{\alpha x}) \exp(-xt)\,\mathrm{d} x, $$ and $${1\over \pi} ...
0
votes
0answers
26 views

Orthogonality of Spherical Bessel Function

How to prove Orthogonality of Spherical Bessel Function?
20
votes
2answers
341 views

Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,dx$ have a closed form?

Mathematica gives an approximate result of $1.581949621806183890451628...$, but no exact form. I predict it's a function of $e$ and $\pi$, and perhaps even the Golden Ratio $\phi$ (It certainly ...
3
votes
0answers
49 views

Double integral of symmetric polylogarithmic function over rectangular region

This question was inspired by M.N.C.E.'s wonderful response here. While exploring the possibility of generalizing his result, I found that a significant part of the problem reduced to evaluating the ...
0
votes
0answers
45 views

Some expectations of psi (digamma) function

I want to derive an Expectation-Maximization algorithm for my model. But some expectations of psi (digamma) function is needed in the procedure. Assuming I have a Gamma distributed random Variable ...
20
votes
2answers
480 views
16
votes
0answers
194 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}\land0<a\le1\land0\le b$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
4
votes
0answers
35 views

Identification of a function

I recently came across the following function $$\sum_{k=1}^\infty(\log(k))^n\frac{z^k}{k}$$ I found it while dealing with the polylogarithm function, $Li_n (z)$ (Notice that if instead of ...
3
votes
0answers
57 views

Integral involving Whittaker function

Consider the following integral: $$ \int_1^{\infty} \frac{e^{u/2}}{u}[-\mathrm{Ei}(-u)]\,W_{1,\imath p}(u)\,du, $$ where $\imath=\sqrt{-1}$ and $p>0$ selected so that $W_{1,\imath p}(1)=0$; here ...
0
votes
0answers
17 views

Finding zeros of a function involving Gamma function.

I am looking for the zeros of following function ($a$ and $b$ are real): $$ F(a,b) = 4^{a+ib} \Gamma(a+ib) \Gamma(-a) \Gamma(-ib) + \Gamma(-a-ib) \Gamma(ib)\Gamma(a) $$ and I have no idea on ...
12
votes
1answer
163 views

Evaluate the double sum $\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2}$

As a follow up of this nice question I am interested in $$ S_1=\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2} $$ Furthermore, I would be also very grateful for a solution ...
4
votes
0answers
106 views

Solving integral with spherical bessel functions

I would like to find if possible a solution (closed form) for the following integral: $$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,j_{0}(b\cos x)\,j_{0}(b\sin ...
4
votes
0answers
73 views

integrate $\int \frac{1}{e^{x}+e^{ax}+e^{a^{2}x}} \, dx$

I've been trying to integrate $$ \int \frac{1}{e^{x}+e^{\omega x}+e^{\omega^{2}x}} \, dx $$ where $\omega=e^{2i\pi/3}$ but to no avail. I've tried substituting in $u=e^{(1+\omega)x}$ but ended up ...
7
votes
6answers
206 views

Solve the following equation: $\sqrt {x + \sqrt {4x + \sqrt {16x + \sqrt {64x + 5}}}} - \sqrt x= 1$

A past examination paper had the following question that I found interesting. I tried having a go at it but haven't come around with any solutions. How would one go about tackling it? $$\sqrt {x + ...
2
votes
2answers
39 views

Finding all values of $\theta$ which describes a straight line

I am having quite a bit of trouble understanding the below question; my assumption is that I should bring the right-hand side in terms of $\sin \theta$ or $\cos \theta$ however am not able to proceed ...
2
votes
0answers
45 views

Conjecture of the general form of a power series

Relcently I met a power series(Source Link-Eq(4.1)) of the type $$ f(x)=1-x+\frac{1}{2}x^2+\frac{1}{4}x^3-\frac{1}{8}x^4-\frac{35}{128}x^5-\frac{157}{1024}x^6+\cdots $$ where $x$ is supposed to be a ...
1
vote
1answer
60 views

the roots & the limit of $2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$

If $$2^{(2\pi)^{\cos(2\pi)}}\sqrt{\cos(2\pi)}=2^{2\pi}$$ Can you obtain or is it plausible to find the roots and the limit of $$2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$$ if $0 < \cos(x)$ and $0 < ...
12
votes
1answer
147 views

The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
3
votes
0answers
55 views

What is $\int \frac{e^{a x}}{1+x^2} dx $?

In my answer to another question (here: Upper and lower bound on different of ${\rm erf}(\frac{x+c}{b})-{\rm erf}(\frac{x-c}{b})$), I came up with this integral: $\int \frac{e^{a x}}{1+x^2} dx $. I ...
2
votes
1answer
25 views

Simplification of Hankel functions

I have this Hankel function, $H_{1}(R_{1}+R_{2})e^{i\cos(a)}$. Would it be possible to simplify this function in terms of $H_{1}(R_{1})$ and $H_{1}(R_{2})$?
1
vote
1answer
84 views

transforming ordinary generating function into exponential generating function

I have seen a post here that says that you can convert an exponential generating function into an ordinary one with the aid of the Laplace transform. Is it possible to do the reverse transformation? ...
0
votes
0answers
21 views

Solve $x$ in the equation: $a\cdot \textrm{arctanh} [b + a \cdot x] - c \cdot \textrm{arctanh} [d + c \cdot x] = e$

How to solve $x$ in the equation: $a\cdot \textrm{arctanh} [b + a \cdot x] - c \cdot \textrm{arctanh} [d + c \cdot x] = e$, where $\textrm{arctanh}(x) = \frac{1}{2} \log \left(\frac{1+x}{1-x} ...
1
vote
1answer
64 views

Gamma and Beta function proof.

I'm trying to proof the equality $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ when $x,y>0,$ without using calculus in many variables. I've investigated about the topic but all references make ...
1
vote
0answers
33 views

What does subcopula mean?

In copula concept, what does "subcopula" exactly mean? Does it mean a subset of copula? Would you please explain a little bit in details? Thanks in advance!
5
votes
0answers
144 views

Can these integrals be represented in closed form?

This paper in the formula F.3.6 (page 271) gives the following formula for the derivative of Hurwitz Zeta function: $$\frac ...
0
votes
1answer
19 views

Infinite Integral of a Bessel Function

I need to calculate the following integral $$ \int_0^{\infty}xdxJ_n(kx) $$ Integrating it by parts and using the normalization of Bessel functions, I find it (somewhat heuristically) to equal the ...
3
votes
1answer
80 views

Concerning Hurwitz Zeta function, how to prove the following identity?

It is claimed that $$\zeta'(0,s)=\ln\left(\frac{\Gamma(s)}{\sqrt{2\pi}}\right)$$ where the derivative is meant by the first argument (as usual with Hurwitz Zeta). How to prove this? Wolfram Alpha ...
1
vote
0answers
22 views

Asymptotic for Bessel Function

We have that, $$J_p(x) = \sqrt{\frac{2}{\pi x}} \sin \left( x - \frac{p\pi}{2} + \frac{\pi}{4}\right) + \frac{r_p(x)}{x\sqrt{x}}$$ We also know that there exists $M>0$ such that $|r_p(x)| \leq M$. ...
1
vote
1answer
40 views

Help Obtaining Numerical Approximation of Lambert W Solution

I am studying a particular generating function $$\frac{2e^x}{e^{2x}+1+2x}$$ and I thought I would try to solve the equation $$e^{2x}+1+2x=0$$ to determine for what value of $x$ if any the function ...
0
votes
0answers
17 views

Orthogonality of Hankel functions, what are the relations?

What are the relations for orthogonality of Hankel's relations defined via: $H^{(1)}_{m}(z) \equiv J_{n}(z) + i Y_{n}(z)$ $H^{(2)}_{m}(z) \equiv J_{n}(z) - i Y_{n}(z)$ I have looked at some books ...
4
votes
1answer
43 views

On Lamda function

The Lambda function is defined as: $$\lambda(s)=\sum_{n=0}^{\infty} \frac{1}{(2n+1)^s},\; \mathfrak{Re}(s)>1$$ How to prove that $\lambda(s)=(1-2^{-s})\zeta(s)$? Basically, I was dealing with ...
1
vote
0answers
40 views

Integrating a product of to error functions and an exponential

I have the following integral that I need to solve. $\int_{-\infty}^\infty \exp(-\frac{x^2}{2})*\text{erf}(x-\delta)*\text{erf}(x-\gamma)dx$ I was hoping I could use this: Integral of product of ...
2
votes
1answer
22 views

$x^2y''+(2x^2+x)y'+(2x^2+x)y=0$ A Bessel equation

$$x^2y''+(2x^2+x)y'+(2x^2+x)y=0$$ The solution is $$e^{-x}J_o(x)+e^{-x}Y_o(x)$$ How does one approach a problem like this?
7
votes
2answers
168 views

Solving $\ln{x}=\tan{x}$ with infinitely many solutions

Lets take $f(x)=\ln{x}$ and $g(x)=\tan{x}$ When $f(x)=g(x)$ that is $\ln{x}=\tan{x}$, we see that the graph is like: Hence we see that there are infinitely many solutions to $x$ but the two ...
1
vote
1answer
59 views

prove an identity involving beta function and gamma function

We know that $B(p,q)=\Gamma(p)\Gamma(q)/\Gamma(p+q)$ where $p, q>0$, and $B(p,q)$ is related to binomial coefficients if one of $p,q$ is an integer. I want to prove the following identity. ...
2
votes
1answer
45 views

elliptic curve isogeny class 14.a $L$-function Dirichlet coefficients

Are the Dirichlet coefficients $a(n)$ of the $L$-function associated with isogeny class 14.a the irrationals that the inverse symbolic calculator suggests they are? The Lcalcfile suggests that they ...
0
votes
2answers
78 views

Prove that $f^{-1} (F)$ is closed

A set $F \subset \mathbb R$ is closed if for any convergent sequence $\{x_n\}$ in F converges, we have $\lim_{n \to \infty} x_n=x \in F $. How to Prove that if $f :\mathbb R \to \mathbb R$ is ...
1
vote
1answer
38 views

Integral over product of two bessel functions and power

I have searched the literature on integrals over bessel functions, but I couldn't find anything. The integral to be evaluated is, $\int_0^a J_{n}(bx)J_{\mu}(cx)xdx =: \mathcal{M}_r(a;n,\mu;b,c)$ ...
6
votes
3answers
136 views

How to solve $x^2 = e^x$

The question is to find $x$ in: \begin{equation*} x^2=e^x \end{equation*} I know Newton's method and hence could find the approx as $x\approx -0.7034674225$ from \begin{equation*} ...
1
vote
1answer
35 views

Integration of the incomplete beta function

I would like to know if there is a way of computing the following integral analytically ($B_u$ is the incomplete beta function): $$\int B_u(a-1,0)~u^{-a} du$$ Thanks for your ideas.
2
votes
1answer
63 views

Closed-form expression for $\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$?

As per the title, I am looking for a closed-form expression for the integral $$\frac{1}{B(\alpha,\beta)}\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$$ where $a,\alpha,\beta>0$ and ...
2
votes
1answer
53 views

Laplace transform involving the gamma function.

Does anyone know how to evaluate the following integral $$ \int_{0}^{\infty} \frac{e^{-qs}\alpha^{s}}{\Gamma(s)\Gamma(s)}\text{d}s $$ where $q,\,\alpha > 0$? I've done some digging in usual ...
1
vote
1answer
87 views

Integrating Associated Legendre Polynomials

As part of a derivation for the question I asked here in Physics stackexchange, I am trying to calculate the following integral, but I am not sure how to proceed: ...
0
votes
0answers
72 views

Handling complex arguments of elliptic integrals in Maple

I want to solve the integral \begin{equation} I(y)=\int_0^y\sqrt{\dfrac{x(1+ax)}{(1+ax)^2-b^2}}\,\mathrm{d}x,\qquad a<0,\; b\in(0,1),\; y>0,\; (1+ax)^2-b^2>0, \end{equation} using Maple 18. ...
2
votes
2answers
54 views

Why are there four independent solutions of Mathieu equation instead of two?

Consider Mathieu equation: $$\frac{d^2}{d\xi^2}R(\xi)+(a-2q\cos(2\xi))R(\xi)=0.$$ It's a second order ODE, so it should have two linearly independent solutions. One of the choices is to denote one ...
3
votes
1answer
85 views

A possible dilogarithm identity?

I'm curious to find out if the sum can be expressed in some known constants. What do you think about that? Is it feasible? Have you met it before? $$2 ...
4
votes
1answer
67 views

Is the solution of functional equation $x^x=y^y$ when $0\lt x\lt y$ uncountable?

I want to prove that, the set: $S=\{(x,y)\in \mathbb R^2\,\,|\,\,0\lt x\lt y \,\,,\,\,\,\,x^x=y^y \}$ $\,\,$ is uncountable. My idea is the following: Consider the function ...
0
votes
2answers
43 views

Writing $f(x)$ in terms of the heaviside function

I have $f(x,t) = 0$ when $t \le 0$ and $f(x,t) = \sin(-x + t)$ when $t > 0$. I have been told this can be written more concisely in terms of the heavisdie function $u(t - a)$ as $f(x,t) = \sin(-x ...