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
54 views

Two properties about Bessel function

Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...
2
votes
1answer
71 views

integral of modified bessel function of 2nd type

I need some help on a possible way to integrate this: $$ \int_0^\infty{x^{m-1}\mathrm{e}^{-\lambda x}\left[\frac{\operatorname{K}_\nu\left(b\sqrt{\alpha+\beta x}\right)}{\left(b\sqrt{\alpha+\beta ...
3
votes
2answers
127 views

Two series involving the Gamma function

The last piece I am left with in my proof is to compute the following two series: $$\sum_{i=1}^{n-1}\dfrac{\Gamma(i-d)\Gamma(n-i+d)}{\Gamma(i+1)\Gamma(n-i)(n-i-d)}, \sum_{i=1}^{n-1} ...
0
votes
1answer
114 views

Complementary error function in matlab

Please I really want to know how to verify the following relation in MatLAB $\text{erfc}(x)\overset{x\rightarrow\infty}{\longrightarrow}\dfrac{e^{-x^2}}{x\sqrt{\pi}}$
2
votes
0answers
47 views

Renormalizing Legendre polynomials to $P_n(0)=1$

One way to define the Legendre polynomials is with the recurrence relation $$(n+1)P_{n+1} (x) = (2n+1)xP_{n} (x)-nP_{n-1} (x),$$ with $P_0(x)=1$ and $P_1(x)=x$. This standardization is normalized so ...
12
votes
2answers
138 views

Calculate the following integral $\int_0^{\pi/2} \frac{\sin^m x\,\mathrm{d}x}{\sin x + \cos x}$, $m=2k-1$

At the moment I am studing the following integral $$K(m,n)= \int_0^{\pi/2} \frac{\sin^m x\,\mathrm{d}x}{\sin^nx + \cos^nx}.$$ For integers $m$,$n$. The question regarding both $K(1,1)$ and ...
24
votes
5answers
401 views

Other interesting consequences of $d=163$?

Question: Any other interesting consequences of $d=163$ having class number $h(-d)=1$ aside from the list below? Let $\tau = \tfrac{1+\sqrt{-163}}{2}$. We have (see notes at end of list), ...
3
votes
0answers
40 views

Choosing an appropriate complete orthogonal basis

I have a function $f(x)$ which I want to represent as the sum over some complete orthogonal basis $\phi_i$ such that: $$ f(x) = \sum_{i} c_i \phi_i(x) $$ Where the $\phi_i$ are orthogonal with ...
0
votes
1answer
46 views

Show that these identities are true.

$\dfrac{\text{d}}{\text{d}x}[x^{-n}J_n(x)]=-x^{-n}J_{n+1}(x)$ $\dfrac{\text{d}}{\text{d}x}[x^{n}J_n(x)]=x^{n}J_{n-1}(x)$ $xJ_n'(x)=nJ_n(x)-xJ_{n+1}(x)$ Where ...
6
votes
1answer
179 views

Why does $\lim_{x \rightarrow 0} B(x,y)$ exist and how is it calculated?

In evaluating integrals like (link to another example) $$I=\int_0^1\frac{\log(x) \log^2(1-x)dx}{x}$$ one can make the substitution $x=\sin^2(\theta)$ to obtain ...
1
vote
1answer
67 views

How prove this $f(x,y)=\dfrac{1}{2\pi}\ln{\dfrac{1}{|x-y|}}+\dfrac{i}{4}-\dfrac{1}{2\pi}\ln{\dfrac{k}{2}}-\dfrac{C}{2\pi}$

let $$J_{0}(x)=\sum_{p=0}^{\infty}\dfrac{(-1)^p}{(p!)^2}\left(\dfrac{x}{2}\right)^{2p}$$ and ...
8
votes
3answers
238 views

How prove this limit $\lim_{\alpha\to n}\dfrac{J_{\alpha}(x)\cos{(\alpha \pi)}-J_{-\alpha}(x)}{\sin{\alpha\pi}}$

let $$J_{\alpha}(x)=\sum_{m=0}^{\infty}\dfrac{(-1)^m}{m!\Gamma{(m+\alpha+1)}}\left(\dfrac{x}{2}\right)^{2m+\alpha}$$ show that: \begin{align*}&\lim_{\alpha\to n}\dfrac{J_{\alpha}(x)\cos{(\alpha ...
3
votes
3answers
187 views

Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
1
vote
1answer
188 views

Evaluate the Bessel Function $J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$

I need to evaluate the following definite integral: $$J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$$ I have attempted basic variable substitution and expanding the cosine term, but I have ...
1
vote
1answer
59 views

How to evaluate the integral involving two Bessel functions and following elementary function?

Let's have the integral $$ \int \limits_{0}^{\infty} J_{0}(kr) J_{0}(kr')\frac{kdk}{k^2 + a^2}. $$ How to evaluate it? I failed when was trying using integral representations for the Bessel ...
1
vote
1answer
25 views

Lipschitz bump function

Does anyone know of an example of a Lipschitz or Holder continuous bump function on $\mathbb{R}^n$? Any help is appreciated. Thank you.
1
vote
0answers
34 views

Norm of a matrix exponential

Can any one prove the following inequality $$||e^{Pt}||\leq e^{t\alpha{(P)}}\sum_{k=0}^{r-1}\frac{(||P||\sqrt{r}\,t)^k}{k!}$$, where $r$ is the order of the matrix $P$ and $\alpha(P)$ be the maximum ...
1
vote
0answers
29 views

parabolic cylinder function with negative argument

Say I wish to calculate $U(0,-1)$ which is $D_{-1/2}(-1)$ using the whitaker notation. According to http://dlmf.nist.gov/12.7.10, the identity is ...
11
votes
1answer
120 views

An identity involving integrals of the form $\int_{0}^{\alpha} \arctan (\sqrt{5}\tan\theta) \, d\theta$.

I want to establish the following identity $$ \int_0^{\pi/3} \arctan (\sqrt{5} \tan \theta) \, d\theta - 2 \int_0^{\pi/6} \arctan (\sqrt{5} \tan \theta) \, d\theta = \frac{\pi^{2}}{30}. \tag{1} $$ ...
39
votes
2answers
395 views

Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$

Using a numerical search on my computer I discovered the following inequality: $$\left|\,{_2F_1}\left(\frac14,\frac34;\,\frac23;\,\frac13\right)-\rho\,\right|<10^{-20000},\tag1$$ where $\rho$ is ...
0
votes
0answers
52 views

Fox H-function and the Riemnann zeta function

What is the link between the Fox's H-function and the Riemnann zeta function or the polylogarithmic function? PS: I would be glad if someone could provide me references about this gadget ...
6
votes
1answer
345 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
1
vote
1answer
33 views

About large z behavior of hypergeometric function $_2F_1(1/2,1/2,1;z)$

The hypergeometric function $_2F_1(\large \frac{1}{2},\frac{1}{2},1;\frac{1-\frac{u}{\Lambda^2}} {2} \large)$ at large $\mid u\mid$ can be approximated by $$ -\frac{\Lambda}{\pi} \sqrt{\frac{2}{u}} ...
1
vote
1answer
73 views

Bessel function with complex argument

So I understand that the bessel functions of the first kind are the ones that satisfy this equation: $$x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-\alpha^2)y = 0$$ and the result is a linear ...
2
votes
1answer
145 views

Evaluating $\int_{0}^{\frac{\pi}{2}} \arctan( a \sin x) \ dx$ using the Taylor expansion of $\arctan x$

I was wondering if it's possible to show that $$\int_{0}^{\pi/ 2} \arctan (a \sin x) dx = 2 \sum_{k=0}^{\infty} \frac{(\frac{(\sqrt{1+a^{2}}-1}{a})^{2k+1}}{(2k+1)^{2}} = 2 \ \chi_{2} ...
0
votes
0answers
38 views

Does $a x+b=\cos(x)$ have a special-functions solution analogous to the Lambert W function?

The Lambert W function is defined as the solution to the equation $z=w e^w$, in the sense that for all $z\in\mathbb C\setminus(-\infty,-1/e]$ we can find a complex number $W(z)$ which obeys ...
0
votes
0answers
25 views

higher order hash functions

I am trying to find a functions which can efficiently transform (input) N number to and Integer K (and uniformly distribute it over some range - will use this as hash function). It is kind of a hash ...
25
votes
1answer
305 views

Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers?

Are (integers) plus (elementary functions) plus (generalized hypergeometric functions) sufficient to represent any algebraic number? For example, the real algebraic number $\alpha\in(-1,0)$ ...
1
vote
1answer
77 views

Solving Some Transcendental Equations

How do you solve for $a$ in each of the equations $$a^{a^a}=b^c$$ $$a^{a^{a^a}}=b^c$$ $$a^{a^{a^a}}=b^{c^d}?$$
28
votes
1answer
371 views

Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

The following integral \begin{align*} \int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96} \tag{1} \end{align*} is called the Ahmed's integral ...
0
votes
0answers
62 views

Definite Integral of Modified Bessel function representation

I am trying to express the following integral of the Modified Bessel function either in closed form or even using other special functions. Any ideas ? $$ \int_{0}^{b}x\exp\left(-\,{x^{2} + z^{2} ...
2
votes
0answers
37 views

How to speedup evaluation of hypergeometric ${}_3 F_2(1)$?

I need to make a table of ${}_3 F_2\left(\frac{a}2+\frac14, \frac{a}2+\frac34, \frac{a+b}2;\; a+1,\frac{a+b}2+1;\;1\right)$ for integer $a, b,$ $0\le a\le N_1$, $0\le b\le N_2$, with precision of 50 ...
18
votes
1answer
288 views

Derivative of the Meijer G-function with respect to one of its parameters

Are there any approaches that allow to find a derivative of the Meijer G-function with respect to one of its parameters in a closed form (or at least numerically with a high precision and in ...
3
votes
1answer
127 views

Monstrous Moonshine for $M_{24}$?

This is connected to my MO post "Monstrous Moonshine for $M_{24}$ and K3?". In page 44 of this paper, eqn(7.16) and (7.19) yield, ...
20
votes
3answers
351 views

Integral $\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$

I need to evaluate this integral to a high precision: $$\large I=\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$$ Symbolic integration in ...
17
votes
2answers
249 views

A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$

Is it possible to evaluate in a closed form integrals containing a squared hypergeometric function, like in this example? ...
7
votes
1answer
163 views

Simplification of $G_{2,4}^{4,2}\left(\frac18,\frac12\middle|\begin{array}{c}\frac12,\frac12\\0,0,\frac12,\frac12\\\end{array}\right)$

In this post Cleo gives a misterious result containing the following generalized Meijer G-function: ...
5
votes
2answers
248 views

Proving Inequality with the Greatest Integer Function

Show that $$[(m+n)x]+[(m+n)y] \ge [mx+(n-1)y]+[my+(n-1)x]$$ where $m,~n \in \Bbb{N}$ and $0\le x,~y < 1$. I've tried everything for about half a day and still couldn't figure it out. ...
8
votes
0answers
75 views

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. ...
0
votes
2answers
40 views

Definite integral of $e^{-by^2}/(y^2-c^2)$

How to get the definite integral like this? $$\int^a_0 \frac{\exp(-by^2)}{y^2-c^2}dy$$ where $a,b,c$ are parameters and $a>c$. Thanks a lot.
1
vote
0answers
37 views

Intertwiner for $U(n-1) \subset U(n)$

I'm using the notation of Vilenkin and Klimyk, ''Part3: Representations of Lie Groups and Special Functions''', chapter 18. Given an irreducible representation $T_m$ of the complex Lie algebra $U(n)$ ...
0
votes
1answer
341 views

Triangular and other step functions in matlab

As the title indicates, I need help on how to plot a triangular function in Matlab. e.g. $ f(x)=\begin{cases} 1-|x|, & |x|< 0 \\ \\ 0, & \text{otherwise} ...
1
vote
2answers
73 views

Copulas, implication

Let $C$ be a copula function. Prove that $C(t,1-t)=0$ for all $t\in[0,1]$ implies that $C(u,v)=\max(u+v-1,0)$. I think the implication other way around is easy to see, however I can't see why the ...
0
votes
1answer
34 views

Proving that a function is grounded

I need to prove that a function $f:[0,1]\times[0,1]\rightarrow [0,1]$, which is nondecreasing in each variable, is grounded (i.e. that $f(0,y)=0=f(x,0)$ for all $(x,y)$ in $[0,1]\times[0,1]$). ...
1
vote
1answer
79 views

How to use following equation by using Green's function?

Let's have the following equation: $$ u''(r) + \frac{1}{r}u'(r) - \alpha^{2}u(r) = f(r), $$ where $r$ is polar radius. Method of Green's function leads to $$ u''(r) + \frac{1}{r}u'(r) - ...
7
votes
1answer
81 views

Calculating $\text{erf}^{-1}(z)$ for $z\in\mathbb{C}$

All the information I found about inverse error function $\text{erf}^{-1}(z)$ was about $z\in\mathbb{R}$. Also I found some Taylor expansions for it, but as the function is unbounded near $z=\pm1$, ...
0
votes
0answers
77 views

Evaluating integral with Legendre polynomials

I want to do the following integral: $$ I=\int_{-1}^{1} x^n P_{n}(x) \rm{d}x $$ WITHOUT using Rodrigues' formula. I'm required to use $$ P_{n}(x) = \sum_{r=0}^{[n/2]} \frac{(-1)^r (2n-2r)!}{2^n r! ...
7
votes
0answers
162 views

How find this value of $A$?

Question: Let $z\in C$ Find this value $A$,such $$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$ where $i^2=-1$,and $w_{k}(z)$ is Lambert $W$ function:see ...
1
vote
1answer
104 views

2-increasing functions

I'm trying to learn about copulas. Two definitions I've come across are the H-volume of a rectangle, which is defined as $V_H=H(x_2,y_2)-H(x_2,y_1)-H(x_1,y_2)+H(x_1,y_1)$. The function H, whose ...
0
votes
1answer
76 views

Rewriting $\delta(x,y)$ in terms of $\delta(r)$.

On my textbook is written: The function $\tau^{-1} u(x/\tau)$ is a rectangle function of height $\tau^{-1}$ and base $\tau$ and has unit area; as $\tau$ tends to zero a sequence of unit-area pulses ...