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

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10
votes
1answer
99 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
32 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
59 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
119 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
36 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
36 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
39 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 < ...
9
votes
0answers
86 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
45 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
19 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
49 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
17 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
47 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
17 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!
3
votes
0answers
69 views
+50

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
15 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 ...
2
votes
1answer
58 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
21 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 $|r_p(x)| \leq M$. ...
0
votes
1answer
28 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
13 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 ...
3
votes
1answer
37 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
27 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 ...
1
vote
1answer
19 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?
6
votes
2answers
142 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
36 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
41 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
75 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 ...
0
votes
1answer
20 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
113 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
26 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
48 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 ...
1
vote
1answer
32 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 ...
0
votes
0answers
30 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
50 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
42 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 ...
2
votes
0answers
54 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
64 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
40 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 ...
0
votes
1answer
27 views

Why it is necessary for copula functions to be grounded?

I know what the properties "grounded" and "2-increasing" means in copula functions definition but actually I can't understand the reason behind these two! I mean why it is necessary for copulas to be ...
2
votes
1answer
24 views

Is this function defined in terms of elliptic $\mathrm{K}$ integrals even?

Let $R,z > 0$ be positive real constants, and consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $$ f(v) = \frac{1}{\sqrt{(R+v)^2+z^2}}\ \mathrm{K}\!\left( \frac{4 R v}{(R+v)^2+z^2} ...
1
vote
0answers
27 views

Proof for the Rodrigues formula for Neumann's Spherical functions.

I've been trying to prove the Rodrigues formula for Neumann's Spherical functions. The Neumann's Spherical functions are: $$N_n(x))=-(-x)^n\left[\frac{1}{x}\frac{d}{dx}\right]^n\frac{\cos{(x)}}{x}$$ ...
5
votes
2answers
229 views

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ ...
0
votes
0answers
13 views

$K$ which is of second category in itself.let $H = K \cap ( - K)$. Why $H$ is non empty interior

Let $X$ be topological vector space.Let $K$ be closed, convex, dense subset of $X$ and $K$ which is of second category in itself. Put $H = K \cap ( - K)$. Why does $H$ is nonempty interior?
1
vote
2answers
66 views

A special modular function: $ j $-invariant.

It is known that j invariant $$j(\tau)= 1728 \frac{g_2^3(\tau)}{\Delta(\tau)} $$ $\tau \in \mathbb{H}$ attains every complex value , Can someone guide me its proof.?? where $L(\tau ) = \{\tau m ...
1
vote
1answer
43 views

Heaviside function & Integral Limits

When considering integration, how does one use the Heaviside function in order to alter the limits of integration. For example If i have $$ \int_a^b f(x) dx $$ But want to change this integral to be ...
2
votes
0answers
28 views

Barnes' double gamma function versus q-gamma function

According to wikipedia, the q-analog of the gamma function is closely related to a multiple gamma function defined by Barnes. Besides the fact that they are both generalizations of the Gamma function, ...
0
votes
1answer
19 views

Difference Between Lyapunov and Strong Lyapunov Function.

Good Day everyone. I was assigned to show that given an autonomous system of Differential Equations and a function $V$, I need to show that $V$ is Lyapunov function. To show that $V$ is Lyapunov. I ...
2
votes
1answer
73 views

Name of function $(1+x)^n-1$

Is there any name for this formula $$(1+x)^n-1$$ When working with floating point numbers this can be calculated with much better precision for very small $|x|<1$ values using Taylor series ...
1
vote
1answer
67 views

How to prove this problem about supermodularity function?

The problem is as follows, and I have solved the subproblem (a), but haven't solved (b) yet. And for (b) the method I think about is proof by contradiction, but I get stuck before I could solve this. ...
3
votes
2answers
49 views

Derivative of the Gamma function

How do you prove that $$ \Gamma'(1)=-\gamma, $$ where $\gamma$ is the Euler-Mascheroni constant?