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

learn more… | top users | synonyms

1
vote
0answers
45 views

How to solve a homogeneous Fredholm integral equation of the second kind with a symmetric non-seperable kernel?

I have the equation \begin{eqnarray} \lambda L(p)=\int dq\,K(p,q)L(q) \end{eqnarray} Where $L$ is an unknown function, $\lambda$ is some constant, and $K$ is a known function. This is a homogeneous ...
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 ...
3
votes
3answers
137 views

What is the inverse function of $e^x +x$?

As the natural $\log(x)$ function is the inverse of the exponential $e^x$ and $\log(x +1)$ is the inverse of $e^x - 1$, what it the inverse of $e^x + x$?
6
votes
0answers
85 views

On $\sum a^n \tan(n\theta)$

It is well known that $$\sum_{n=0}^{\infty} a^n \cos(n\theta) = \frac{1-a\cos(\theta)}{1-2a\cos(\theta)+a^2}$$ $$\sum_{n=0}^{\infty} a^n \sin(n\theta) = \frac{a\sin(\theta)}{1-2a\cos(\theta)+a^2}$$ ...
1
vote
1answer
30 views

On A Relation of the Gamma Function to a Certain Condition

I have researched to find an answer to this question to no avail. Does $$\Gamma \left( s \right) =- \int\limits_0^\infty \frac{t^{s - 1}}{(e^{2t}-e^{t})}\,dt$$ iff $Re(s) = \frac 1 2$? (Where $\Gamma ...
0
votes
0answers
18 views

Number of possible unate functions possible

An unate function f is one which is constant or can be represented by an SOP using either complemented or uncomplemented literals for each variable. My question is : How many such unate functions in ...
2
votes
0answers
22 views

Bessel Function Integral $\int ^{2 \pi}_0 e^{i x \cos t + n t}dt=2\pi i^nJ_n(x),n\in\mathbb{Z}$

$$\int ^{2 \pi}_0 e^{i x \cos t + n t}dt=2\pi i^nJ_n(x),n\in\mathbb{Z}$$ This holds for integer n (although I do not understand why), but what is it equal to if n is not an integer?
1
vote
4answers
106 views

What is the function that satisfies $\int_0^x f(t) dt=constant$ [closed]

$$\int_0^x f(t) dt=constant$$ What is the function that satisfies this condition ? Thank you!
2
votes
3answers
660 views

What does | mean?

I found this symbol on Wolfram|Alpha. Does it mean "or"? $\displaystyle \large \cos^{-1}(-1)=\mathrm{cd}^{-1}(-1\mid 0)$
1
vote
0answers
35 views

Forming the Differential Equation from the given Solution

The solution of a Differential equation is given as: $Y(x)= x^nJ_n(x)$ where $J_n(x)$ is Bessel's function(of first kind) of order $n$. I wanted to find the Corresponding Differential Equation.
1
vote
0answers
21 views

Question about finding a function $f(x)$ that fits the following bounds

I am in search of a function $f(x)$ that is infinetely differentiable (that is $f^{(n)}(x) \ne 0$ and that the function is defined when differentiated $n$ times) and an interval $[a,b]$ that will ...
0
votes
1answer
27 views

An exponential upper bound for Bessel K function?

I saw a bound of the form: $$K_\nu(x) \leq Ce^{-x} \quad\text{for $x \geq 1$}$$ i.e. an exponential bound, somewhere, but I have no reference. Could someone tell me if this true?
3
votes
1answer
76 views

Asymptotic expansion of Bessel Function

Hi I am interested in calculating an asymptotic expansion of the following function. Or, I would at least like to know how the function behaves for large values of x. I am having trouble simplifying ...
0
votes
1answer
21 views

Bessel K function upper bound $K_\nu(x) \leq \frac{1}{x^{\nu}}$

Let $x, \nu \geq 0$. By WolframAlpha I found that $$K_\nu(x) \leq \frac{1}{x^{\nu}}$$ is an upper bound. I want to know, can this upper bound be improved? Where can I find such properties? Thanks
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 ...
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 ...
0
votes
1answer
98 views

Radioactive Decay formula is $A=A_0e^{-kt}$. How many years until 10 grams decay so that only 8 remain

I have been trying this question for hours and come to a dead end every time... Consider the radioactive decay formula $A=A_0e^{-kt}$ where $A$ is the amount of radium remaining at the time $t$. ...
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 ...
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... ?
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 ...
8
votes
0answers
309 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 ...
5
votes
1answer
113 views

Rogers-Ramanujan continued fraction in terms of theta functions?

The Rogers-Ramanujan cfrac is, $$r = r(\tau)= \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\ddots}}}$$ If $q = \exp(2\pi i \tau)$, then it is known that, $$\frac{1}{r}-r ...
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
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 ...
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
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 $\bar{u}(s,x) = e^{2s(x-t)}$, $s > 0$.
2
votes
1answer
71 views

Solution of second order linear ODE

I consider a second order linear ODE : $$ x^{2\beta+2}\frac{\partial^2 V}{\partial x^2}+(a+x^{2\beta})x\frac{\partial V}{\partial x}+(b+x^{2\beta})V=0. $$ I am expecting that the above equation can ...
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 ...
6
votes
3answers
1k views

Does sinc function have any special inverse function defined?

We know that $y=xe^x$ cannot be solved for $x$ using elementary functions. The Lagrange inversion theorem can be used for finding a "new" function that would be the inverse function of the above ...
3
votes
2answers
228 views

Integrals of Hermite polynomials over $(-\infty, 0)$

Does there exist a simple expression for integrals of the form, $I = \int_{-\infty}^0 H_n(u) H_m(u)\, \mathrm{e}^{-u^2}\,du$, where $m$ and $n$ are nonnegative integers and $H_n$ is the $n$'th ...
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 $b$. ...
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 ...
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 $ ; $ ...
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 ...
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 ...
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 ...
14
votes
2answers
259 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 ...
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? ...
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 ...
3
votes
2answers
117 views

Application of Bessel Function

I have read number of books and online literature on Bessel function. Theoretically, I have known about Bessel function. What is practical significance of Bessel function? How can Bessel function ...
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} = ...
1
vote
2answers
3k views

Meaning of function with circle and cross

I've seen this function M2 = tmp ⊕ Pi. What does the circle with cross do?
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 ...
1
vote
1answer
152 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
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 \ ...
16
votes
3answers
490 views

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

I'm trying to find a closed form of this sum: $$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$ WolframAlpha gives a large expressions containing multiple ...