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
1answer
70 views

$\int_0^\infty\frac{K_0(x)K_0(\alpha x)}{K_0(\beta x)}\cos xy\phantom{.}dx$ Integral from 1926 electrotechnical paper

Erdelyi et.al "Table of integral transforms, vol. I" on p. 50 cites the following integral $$ \int_0^\infty\frac{K_0(x)K_0(\alpha x)}{K_0(\beta x)}\cos xy\phantom{.}dx, $$ but instead of printing the ...
0
votes
1answer
34 views

Does the derivative of the Lambert W function's identity still hold equal?

As the title states, I want to differentiate the identity of the Lambert W function. (I have a tendency to use brackets) Identity: $\frac{x}{W(x)}=e^{W(x)}$ If you don't know what the Lambert W ...
1
vote
0answers
28 views

Confluent Hypergeometric Function behaviour when $x \rightarrow \infty$

I'm very new to confluent hypergeometric functions so please bear with me. What I'm trying to prove is that $$M \left (\frac{c+m}{2m}, \frac{1}{2}, \frac{m}{2d}x^2 \right ) \rightarrow \infty \quad ...
3
votes
2answers
75 views

Prove the validity of gamma function equation $\Gamma(n)\Gamma(n+1/2) = 2^{1-2n}\sqrt{\pi}\;\Gamma(2n)$

How to prove this identity for natural $n$? $$\Gamma(n)\Gamma(n+1/2) = 2^{1-2n}\sqrt{\pi}\;\Gamma(2n)$$ Firstly, I set $n=1$ and looked at general gamma equation. How to simplify or... ?
4
votes
1answer
128 views

Prove that $\int_{-\infty}^\infty \frac{\operatorname{Ai}^2(x+a_n)}{x^2}dx = 1$

While I've been thinking about this question, I've found that for all $n \geq 1$ integer values, we have $$ \mathcal{I}_n = \int_{-\infty}^\infty \frac{\operatorname{Ai}^2(x+a_n)}{x^2}dx ...
2
votes
0answers
52 views

Can anyone identify the orthogonal polynomial for this recurrence relation?

I have come across this recurrence relation: \begin{equation} x p_n(x) = (N - n)(n + 1) p_{n+1}(x) + (N - n + 1) n p_{n - 1}(x) \end{equation} with $p_{-1}(x) = p_{N + 1}(x) = 0$. I expect $p_n(x)$ ...
0
votes
1answer
18 views

Upper Bound for indicator function

For a given $t \in \mathbb{R}$, I want to know if there is a tighter bound on the function $u(x) =\mathbb{1}_{(x \geq t)}$ than $\bar{u}(s,x) = e^{2s(x-t)}$, $s > 0$.
0
votes
1answer
19 views

Function with domain $\mathbb{R}$ that is unbounded around any given point?

What is a function with domain $\mathbb{R}$ that is unbounded around any given point? I apologize for my poor translation of the problem. I didn't know the exact words as we don't study math in ...
0
votes
0answers
47 views

$F(a,c)<1$ if $a=c$ and $F(a,c)\geqslant1$ if $a\neq c$

What is a function $F$ (not constructed from step functions) defined for all real or (if possible) complex numbers pairs such that: $F(a,c)<1$ for all $a=c$ and $F(a,c)\geqslant1$ for all $a\neq ...
7
votes
1answer
78 views

Evaluating an integral - is it a two dimensional beta function? This arises from a variant of Goldbach's conjecture.

Let $\gamma>0$. I would like a nice way to prove that $$\int_{\begin{array}{c} 0\leq s,t\leq1\\ s+t\leq1 ...
0
votes
1answer
59 views

How to solve the first order ODE by separation of variables

There is a first-order ODE $$\frac{dy}{dt}=\frac{a(\ln\frac{1-c}{1-y})^3}{\frac{b-y}{1-y}+\ln\frac{1-c}{1-y}},$$ which is subjected to the initial condition $y(t=0)=y_0$ with $a,b,c$ are all ...
3
votes
0answers
40 views

Is $\sum_n\exp(ian+ibn^2+icn^3)$ known in terms of anything else?

For arbitrary $a,b,c$, does the series $$F(a,b,c)=\sum_{n=-\infty}^\infty\exp\left(ian+ibn^2+icn^3\right),$$ i.e. an evenly-weighed series of exponentials of cubic polynomials, converge to anything ...
2
votes
0answers
37 views

Inverse Laplace transform of a hypergeometric function

I managed to solve an initial value problem in the Laplace domain in terms of a special function $ F(s) = c_2 \frac{1}{{{\left( {{s}^{1 +\beta}}-1\right) }^{\frac{1}{\beta+1}}}}+ c_1 ...
1
vote
2answers
47 views

How does this differential equation define an oscillation from a to b?

The differential equation reads: $$ \dot{y}^2+(y^2-a^2)(y^2-b^2)=0 $$ where $a,b\in \mathcal{R}$ and $a<b$. It is claimed that this differential equation defines an oscillation from $a$ to $b$. ...
0
votes
1answer
75 views

Expanding Fourier Series of $f(x)=\pi-x$ where $0<x<\pi$ (even and odd)

Please help me solve this Fourier series and correct my solution if it is wrong. it's a non-periodic function which we need to write its Fourier series (even and odd) : $ f(x)=\pi - x $ ; $ ...
3
votes
0answers
44 views

Inverse Mellin of the exponential of the digamma function

(Cross-posted from mathoverflow: No answers yet; bounty there expires in less than 24 hours) I'm looking for a function $f_p(x)$ with real parameter $p>0$ satisfying $$ \int_0^\infty ...
1
vote
2answers
77 views

Proving $\Gamma (\frac{1}{2}) = \sqrt{\pi}$

There are already proofs out there, however, the problem I am working on requires doing the proof by using the gamma function with $y = \sqrt{2x}$ to show that $\Gamma(\frac{1}{2}) = \sqrt{\pi}$. ...
0
votes
0answers
24 views

Functions like EWMA

I have a problem in which a threshold triggers an event. I want to estimate that threshold dynamically as I get more information progressively. Exponentially Weighted Mean Average is one such ...
3
votes
1answer
153 views
2
votes
1answer
66 views

Closed form for ${\large\int}_0^1x\,\operatorname{li}\!\left(\frac1x\right)\ln^{1/4}\!\left(\frac1x\right)dx$

Let $\operatorname{li}(x)$ denote the logarithmic integral: $$\operatorname{li}(x)=\int_0^x\frac{dt}{\ln t}.$$ How can we prove the following conjectured closed form? ...
5
votes
0answers
85 views

$f(x+1)=f(x)+f(\alpha\cdot x)$

I try to find an analytic increasing function $f_\alpha$ ($0\le\alpha\le1$) from $\mathbb R$ (or $\mathbb R^+$) to $\mathbb R$ such that for all $x$ $$f_\alpha(x+1)=f_\alpha(x)+f_\alpha(\alpha\cdot ...
0
votes
0answers
38 views

Function expression for reverse 'S'-like shape

I need the expression of a function that looks like the following: The expression preferably needs to be simple e.g., comprised of as few elementary functions as possible. It doesn't matter what ...
2
votes
2answers
54 views

What are the first few values of this function?

There exists a sequence $a_n$ such that $a_n$ is strictly positive, decreasing, well defined for all $n \in \mathbb{Z}^+$, and obeys the following relationship: $$\frac{a_n +a_{n+1}}{2} = ...
3
votes
2answers
50 views

Indefinite Bessel integrals

I just ran into integrals of the Bessel type, but which are unfortunately indefinite integrals, such as $$ f(t)=\int \cos(\gamma\cos(\omega t))\cos(\omega t)\mathrm dt. $$ I'm conscious of the fact ...
1
vote
1answer
156 views

Properties of the Gamma function

How I can show that $$\prod_{r=2}^{N}\frac{\Gamma^2(r\alpha+1)}{\Gamma((r-1)\alpha+1)\Gamma((r+1)\alpha+1)} \lt 1$$ for all $N \gt 2$ and $0 \lt α\leq 1$? Thanks
1
vote
1answer
71 views

Help in Solving a linear Partial differential equation

I can not to solve the following equation $$(*) \qquad u''(r) +2n\coth(r)\,u'(r)+ (n^2+\lambda^2) \, u(r)= 0 \quad \mbox{with} \, r>0$$ where $n\in \mathbb N$ and $\lambda \in \mathbb C $. That I ...
0
votes
0answers
33 views

Smallest integer function and modulo multiplicative inverse

We have to find the value of $\left \lceil \frac{a}{b} \right \rceil mod \ m$, where $m$ is always prime number. I know how to calculate $\left ( \frac{a}{b} \right ) mod \ m$ (It is same as ($((a \ ...
1
vote
0answers
42 views

Heaviside step function composed with a function

I am trying to solve the following equation $$\int_{-\infty}^{+\infty} \Theta(g(\tau))f(\tau)d\tau$$ with $\Theta(t)$ being the heaviside step function defined by: $$\left\{\begin{matrix} \Theta (t) ...
4
votes
3answers
106 views

Integration of Legendre Polynomial

I need to evaluate the following integral \begin{equation} \int_{-1}^1 \frac{d^4P_l(x)}{dx^4}P_n(x)dx\end{equation}. Of course the answer I need is in terms of $l$ and $n$. Does anyone have any idea ...
1
vote
1answer
31 views

Function with the property that its derivative minus its square is a constant

For example cot(x) has this property, as does tanh(-x) and coth(-x). I am looking to approximate the following dome-shaped function: $s(x)=\exp(-(x-\theta_1)^2/\theta_2)$ with a function that has ...
1
vote
0answers
41 views

Derivative of Legendre Polynomial

I am given with a relation \begin{equation} \frac{d^2}{dx^2}(P_l(x))=\frac{1}{2}\sum_{n=(0,1),2}^{l-2}(l-n)(l+n+1)(2n+1)P_n(x). \end{equation} The above sum starts with 0 for even end point, and 1 for ...
1
vote
0answers
14 views

How to simplify $s_t=A_tsin (\omega_0t+\eta sin(\omega_mt))$?

$A_tsin (\omega_0t+\eta sin(\omega_mt))$=$$A_t \{J_0(\eta) cos (\omega_0t)-2 sin(\omega_0t)\sum_{n=0}^{\infty}J_{2n+1}(\eta) ...
0
votes
1answer
35 views

How to solve differential equation $y''(x)=y(x)\Big(1-\frac{\omega_1}{x}+\frac{\omega_2}{x^2}\Big)$

How to solve differential equation $$y''(x)=y(x)\Big(1-\frac{\omega_1}{x}+\frac{\omega_2}{x^2}\Big)$$ where $\omega_1, \omega_2$ is the any nonzero complex number.
0
votes
0answers
31 views

Spherical Fourier Transform of $p_t$

Let $p_t$ denote the heat kernel of the Laplacian $\Delta = \sum_{i=1}^{n}\frac{d^2}{dx_{i}^2}$ on $\mathbb R^n$, I want compute the Spherical Fourier Transform of $p_t$ on $\mathbb R^n$ ? Thank you ...
1
vote
1answer
34 views

What is $\operatorname{frac(x)}$ or $\{x\}$?

I understand this is an opinion kind of a question...but still: Well I know that $\operatorname{frac(x)}$ or $\{x\}$ stands for the fractional part of $x$ but how is it exactly defined? Quoting ...
0
votes
1answer
40 views

Fourth derivative of a Bessel function

I am given with a relation \begin{equation} \frac{d^2}{dx^2}(P_l(x))=\frac{1}{2}\sum_{n=(0,1),2}^{l-2}(l-n)(l+n+1)(2n+1)P_n(x). \end{equation} Using the above equation, I get \begin{equation} ...
0
votes
1answer
22 views

Making sense of a strange set to set mapping

Suppose I have a function $f$ defined as $$f(y) = \{z \in [0, 1] \mid (ay-c) z \leq (ay - c)x\}$$ Where $y \in [0, 1], x \in [0, 1]$ and $ a, c \in \mathbb{R}$ Then claim: $f(z) = 0$ if $(ay-c) ...
0
votes
1answer
41 views

How to prove an expression including Bessel function is positive?

The modified Bessel function of order n of the first kind is given by $$I_n(x)=\sum_{m=0}^{\infty}\frac{(\frac{1}{2}x)^{2m+n}}{m!\Gamma(m+n+1)}$$ where $\Gamma$ is defined by an improper integral, ...
0
votes
0answers
23 views

How to show the order equality of modified Bessel function?

The modified Bessel function of order n of the first kind is given by $$I_n(x)=\sum_{m=0}^{\infty}\frac{(\frac{1}{2}x)^{2m+n}}{m!\Gamma(m+n+1)}$$ where $\Gamma$ is defined by an improper integral, ...
0
votes
1answer
86 views

A nice form of a given function

First let, $\oplus(a_1,a_2,\ldots,a_n)$ denote the bitwise xor of $a_1,a_2,\ldots,a_n$. Define the function $\Delta(a_1,a_2,\ldots,a_n)$ to be the maximum value of $a_i - ...
0
votes
3answers
39 views

Prove that the following function $f(x)=\log_{2}\left(x+{\sqrt{x^2+1}}\right)$ is invertible on the whole number line.

I think that it would help to show that this function is odd. But how can I show that the function $$\log_{2}\left(x+{\sqrt{x^2+1}}\right)$$ is invertible?
8
votes
4answers
283 views

Another beautiful arctan integral $\int_{1/2}^1 \frac{\arctan\left(\frac{1-x^2}{7 x^2+10x+7}\right)}{1-x^2} \, dx$

Do you think we can express the closed form of the integral below in a very nice and short way? As you already know, your opinions weighs much to me, so I need them! Calculate in closed-form ...
1
vote
1answer
36 views

Expansion coefficients of the constant function for bessel series.

I am interested in the following coefficients that are related to the Fourier expansion of the constant function 1 in a Bessel basis. Define $J_n:\mathbb{R}\to \mathbb{R}$ as the $n$'th order ...
1
vote
1answer
39 views

Problem regarding the connectedness of the Topologist's Sine Curve

$$X=A\cup B=\{(x,\sin({\pi \over x}): 0\lt x\le 1\} \cup \{(0,y) : -1\le y\le 1\}\ \subset \mathbb R^2$$ is called the Topologist's Sine Curve - I . Now what is proved is that $X$ is ...
4
votes
2answers
74 views

Closed form of $\ln^n \tan x\, dx$

Here is an integral I am really stuck at. I am pretty sure that a general closed form of the integral: $$\mathcal{J}=\int_0^{\pi/2} \ln^n \tan x\, {\rm d}x, \;\; n \in \mathbb{N}$$ exists. Well if ...
1
vote
0answers
174 views

How to proof that the gamma function is a special function

how do you prove that the euler gamma function is not a combination of elementary functions? I know the liouville theorem that proofs this fact for special functions defined by an indefinite integral. ...
5
votes
0answers
68 views

Integrate $\int x|J_1(x)|^2 \, dx$

Hi I am trying to integrate: $$ \int_0^R x|J_1(bx)|^2 \, dx,\quad R>0, b\in \mathbb{C} $$ where $|J_1|^2=\bar{J}_1 J_1$ where $\bar{J_1}$ means complex conjugation of the bessel function. My ...
9
votes
1answer
333 views

a conjectured continued-fraction for $\cot\left(\frac{z\pi}{4z+2n}\right)$ that leads to a new limit for $\pi$

In this post,I posed a similar conjecture for $\displaystyle\tan\left(\frac{z\pi}{4z+2n}\right)$ but did not get any helpful answers. Given a complex number ...
8
votes
0answers
310 views

a conjectured continued fraction for $\displaystyle\tan\left(\frac{z\pi}{4z+2n}\right)$

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for ...