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

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1
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1answer
49 views

Proof of $\lim_{x\to\infty}\frac{\text{Ei(x)}}{e^x}=0$

I encountered the following limit while doing calculation $$\lim_{x\to\infty}\frac{\text{Ei(x)}}{e^x}=0$$ which is equivalent to $$\lim_{x \to \infty }e^{-x}\sum_{n=1}^{\infty}\frac{x^n}{n·n!}=0$$ and ...
6
votes
1answer
86 views

Integral representation for Fibonacci's numbers

We know that, for example, the Gamma function is a perfect integral representation for the factorial $n!$ for a natural number $n$. $$\Gamma[n] = \int_0^{+\infty} t^{n-1}e^{-t}\text{d}t = (n-1)!$$ ...
7
votes
1answer
149 views

The elliptic integral $\frac{K'}{K}=\sqrt{2}-1$ is known in closed form?

Has anybody computed in closed form the elliptic integral of the first kind $K(k)$ when $\frac{K'}{K}=\sqrt{2}-1$? I tried to search the literature, but nothing has turned up. This page ...
1
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1answer
28 views

proof the derivate of gamma function using the limit definition

using $\Gamma(z+1)=z\Gamma(z)$ and $\Gamma(z)=\lim\limits_{n\to+\infty}\frac{n!n^z}{z(z+1)\cdots(z+n)}$ proof that $$\psi(z+1)=-\lim_{n\to\infty}\left(\sum_{m=1}^{n}\frac{1}{m}-\ln ...
3
votes
4answers
46 views

Derivation of a function over $\frac{1}{\sinh(t)}\frac{d }{dt}$

I can not calculate the next derivative, someone has an idea $$\left( \frac{1}{\sinh(t)}\frac{d }{dt} \right)^n \left( e^{z t} \right)$$ Where $n\in \mathbb N$, $t>0$ and $z\in \mathbb C$. Thanks ...
2
votes
0answers
57 views

Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$ It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity ...
1
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1answer
30 views

Integral representation of the modified Bessel function involving $\sinh(t) \sinh(\alpha t)$

I've come across this peculiar integral representation for $K_\alpha(x)$: $\frac{\alpha}{x}K_\alpha(x) = \int_0^\infty dt \sinh(t) \sinh(\alpha t) e^{-x \cosh(t)}$ How does it come about? Are there ...
3
votes
1answer
106 views

Special functions related to $\sum _{n=1}^{\infty } \frac{x^n \log (n!)}{n!}$

While doing some caculation related to von Neumann entropy, I encountered this kind of convergent series. $$\text{Exl}(x) \equiv \sum _{n=1}^{\infty } \frac{x^n \log (n!)}{n!}$$ In my calculation, ...
0
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0answers
26 views

Seeking a closed form solution to an ODE, if any.

I am try to solve the following ODE by the method of separation of variables. $$\frac{dF}{dx}=\frac{b}{x\left\{\ln^3 x+\ln x[c+d(ax+b)][2\ln x+c+d(ax+b)] \right\}},$$ where $a,b,c,d$ are constants. I ...
1
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0answers
37 views

Existence of solution for equation with erfc

I'm looking for the self-consistent (e.g. input needs to be the same as output) solution of $r$ in $$ r = \frac{1}{2}\operatorname{erfc}(z(r)) $$ where $\operatorname{erfc}$ is complimentary error ...
0
votes
0answers
10 views

Resolvent kernel of Hyperbolic space

The expression of the spherical functions $\varphi_\lambda(x)$ and the resolvent kernel $r_\lambda(x)$ on Hyperbolic space are known. The spherical functions are the Jacobi functions ...
0
votes
0answers
39 views

Relation between hypergeometric functions $_2F_1$ of $z$ and $\frac{1}{1+z}$.

What are the relation between hypergeometric functions $_2F_1$ of $z$ and $\frac{1}{1+z}$. Specifically, I need a transformation that transforms: $_2F_1\left(a,b;c; -\sinh^2(x)\right)$ to ...
5
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3answers
63 views

Alternative definition of Gamma function. Show that $ \lim_{n \to \infty} \frac{n! \; n^m}{m \times (m+1) \times \dots \times (m+n)} = (m-1)!$

Alternative definition of Gamma function on Wikipedia has it defined as a limit. $$ \Gamma(t) = \lim_{n \to \infty} \frac{n! \; n^t}{t \times (t+1) \times \dots \times (t+n)}$$ How do we recover ...
1
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1answer
72 views

Relation between these expressions involving the Hypergeometric function and the Gegenbauer polynomials

I would like to find the relation between the solutions of a differential equation obtained by two different authors. The first solution is given in terms of the hypergeometric function $_2F_1$: ...
9
votes
1answer
137 views

How do I develop numerical routines for the evaluation of my own special functions?

This question has been cross-posted to ComputationalScience.SE here. When performing computational work, I often come across a univariate function, defined in terms of an integral or differential ...
2
votes
1answer
67 views

Product of two hypergeometric functions

For $\Re a, \Re b, \Re c, \Re a', \Re b', \Re c'>0$, I would calculate the following product $$ {}_2 F_1(a, b; c; x^{-1}) \times \, {}_2 F_1(a', b'; c'; 1-\frac{x}{y}) $$ For all $y>x>1$. ...
2
votes
1answer
104 views

meijer g function explicit form

Can the following case of the Meijer G-function $$ G_{2,3}^{3,1}\left(z\left|\begin{smallmatrix}0,1\\ 0,0,0\end{smallmatrix}\right.\right) $$ be expressed more explicitly (in terms of other special ...
6
votes
1answer
125 views

How to prove that only the sine waves keep their shape when they are added together and have the same period?

If $f(t)$ is periodic and $f(t) + C \cdot f(t + t_1)$ has the same shape of $f(t)$ for each value of $C$ and $t_1$, then $f(t)$ has the shape of a sine wave. Is there a simple proof? Is there an ...
3
votes
2answers
145 views

The trigonometric solution to the solvable DeMoivre quintic?

Using the relations for the Rogers-Ramanujan cfrac described in this post, $$\frac{1}{r}-r = x$$ $$\frac{1}{r^5}-r^5 = y$$ and eliminating $r$ yields, $$x^5+5x^3+5x = y$$ This is the case $a=1$ ...
2
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2answers
37 views

Integration of hypergeometric functions?

I would calculate the following integral \begin{equation} I_x = \int_{0}^{1} y^{b+\mu-1} (1-y)^{\nu-1}\, _2F_1(a,b+\nu +\mu;c; xy) \, dy. \end{equation} Such that $\quad \Re a,\Re b,\Re \mu, \Re \nu ...
2
votes
1answer
96 views

Show $\int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x} = \frac{\pi z}{sin(\pi z)}$

I need to solve the following integral: $$ I = \int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x}. $$ Wolfram Alpha gives the answer as $ \frac{\pi z}{sin(\pi z)}$, or equivalently, $\pi ...
0
votes
0answers
77 views

How Can I Find A Closed Form For This Double Summation?

I am looking for a closed form for the following summation that resembles the binomial theorem (to some degree): $$ F_n(x,z) = \sum_{k=2}^n \sum_{c=1}^{k-1} \frac{L_c^{(k-2c)}(-fg)}{(k-c)!} ...
4
votes
1answer
43 views

Fundamental solution of a shifted operator

what is the fundamental solution of the shifted operator $ \Delta + \lambda^2 $, i.e, what the function $f$ satisfying the following equation $$ (\Delta + \lambda^2 )f(x) = \delta(x),$$ where $ \Delta ...
0
votes
0answers
16 views

Zernike Polynomials in Higher Dimensions

Let $n,l$ be integers with $n-l\geq 0$. Set $\alpha =\frac{n-l}{2}$ and $\beta =\frac{n+l}{2}.\ $Then the radial part of the Zernike polynomials in dimension 4 is given by $\tag 1R_{n}^{(l)}(\rho ...
0
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1answer
48 views

How to solve harmonic oscillator-like equation with $\theta$-function?

Suppose second order linear differential equation $$ \frac{d^2 y(t)}{dt^2} + \omega^2(t)y(t) = 0, \quad \omega^2(t) = q^2 + \theta (t-t_{0})m^2 $$ ($\theta (t)$ denotes Heaviside step-function) with ...
0
votes
0answers
36 views

When $\frac{e^{-\lambda } \lambda ^x}{x!}$ over positive integers is invertible?

I am curious for what values of $\lambda \in \mathbb{R}^+$, the function $f(x)=\frac{e^{-\lambda } \lambda ^x}{x!}$ defined only on positive integers i-e $x \in \mathbb{Z}^+$ is invertible? When ...
3
votes
1answer
37 views

Orthogonality of Laguerre polynomials from generating function

I'm trying to show the orthogonality relation for Laguerre Polynomials $L_n(x)$ through their generating function $G(x,t)$. $$G(x,t)=\frac{1}{1-t}e^{\frac{-xt}{1-t}}=\sum_{n=0}^{\infty} L_n(x) t^n$$ ...
2
votes
0answers
44 views

What is known about $\sum_{n=0}^{\infty} x^{n^3} $.

$f(x) =\sum_{n=0}^{\infty} x^{n^2}$ and similar "theta-type" functions are extensively studied. They have many properties and occur in number theory , algebra (in particular solving the quintic ...
2
votes
0answers
30 views

(n+1)-th derivative of the following function [closed]

I have the Runge function $f(x)=\frac{1}{1+x^2}, x\in [-5,5]$ and the $n$-th derivation of this function: $$f^{(n)}(x)=(-1)^nn!\frac{\sin((n+1)\text{arccot}(x))}{(\sqrt{1+x^2})^{n+1}}.$$ I have to ...
2
votes
1answer
61 views

Zeros of Bessel functions

Let $J_\nu(x):=\displaystyle\sum^\infty_{k=0}\frac{(-1)^k(x/2)^{\nu+2k}}{k!~\Gamma(\nu+k+1)}$ denote a Bessel function. When $\nu\geq0$, let $0<j_{\nu,1}<j_{\nu,2}<\cdots$ denote the positive ...
0
votes
0answers
43 views

How can simplify this expression?

Can we write the following expression $$ c_1 P_{i \lambda - \frac{1}{2}}^{\frac{3}{2}}(\cosh t) + c_2 Q_{i \lambda - \frac{1}{2}}^{\frac{3}{2}}(\cosh t); \quad t>0 \, \mbox{and} \, \lambda \in ...
3
votes
1answer
58 views

On what domain is the dilogarithm analytic?

The series $\displaystyle\sum \dfrac{z^n}{n^2}$ converges for $\lvert z\rvert<1$ by the ratio test, meaning that the dilogarithm function $\text{Li}_2(z),$ which is equal to the series ...
1
vote
0answers
56 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 ...
6
votes
0answers
91 views

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

It is well known that $$\sum_{n=0}^{\infty} a^n \cos(n\theta) = \frac{1-a\cos(\theta)}{1-2a\cos(\theta)+a^2}$$ $$\sum_{n=0}^{\infty} a^n \sin(n\theta) = \frac{a\sin(\theta)}{1-2a\cos(\theta)+a^2}$$ ...
3
votes
3answers
138 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 $e^x + x$?
0
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0answers
19 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 ...
1
vote
1answer
31 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 ...
2
votes
0answers
24 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
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1answer
66 views

Solving a differential equation of order two

I can not to solve the following equation $$ y^{ ''}(t) + \left( \lambda^{2} - \frac{2}{\sinh^{2}(t)} \right) y(t) = 0, \quad \mbox{with} \, t>0 $$ where $\lambda \in \mathbb C$. Someone can ...
1
vote
4answers
112 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!
1
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0answers
43 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
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0answers
22 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
29 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?
0
votes
1answer
26 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
89 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
111 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 $t$. ...
2
votes
1answer
73 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
35 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
34 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 ...