Questions tagged [special-functions]

This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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2
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0answers
20 views

On Logarithmic Derivative transformation

I am curious about a certain transformation called the logarithmic derivative that seems to appear a lot in different cool ideas, for example: The use in generating functions for recursions of the ...
0
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21 views

H-function of the exponential function

How could we simplify the following Fox's H-function? $$ H=H_{0\,\,\,1}^{1\,\,\,0}\left[e^{-z}\bigg| \begin{array}{c} - \\ ...
2
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1answer
96 views

Beautiful closed form of $\sum_{n=1}^{\infty}\frac{(-1)^n}{2n^2}\int_0^1\left(\ln f(x)+2\ln g(x)\right)dx$

While solving the following the integral which I found here in brilliant $$\int_0^1\left(\ln(4-3^x)+\ln(1+3^x)\right)dx$$ I happen to create the general integral and variant for the aforementioned ...
4
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1answer
79 views

Proving a formula for $\int_{x=0}^\infty \frac{\sin(ax)x}{(x^2+1)^c} dx$ involving Gamma and Bessel K functions

In Mathematica, $$\int_{0}^\infty \frac{\sin(ax)x}{(x^2+1)^c} dx =\frac{2^{\frac{1}{2}-c}a^{-\frac{1}{2}+c}\pi^{\frac{1}{2}}\operatorname{BesselK}[-\frac{3}{2}+c,a])}{\Gamma[c]} ,$$ where a is a ...
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1answer
38 views

Solving a sum using the hypergeometric function

While answering this question I came across this curious sum: $$S_k=\sum_{n=0}^{\infty}\frac{x^n(n+k+1)!}{n!}$$ Which Wolfram Alpha effortlessly evaluates as $$S_k=\frac{(k+1)!(1-x)^{-k}}{(x-1)^2}$$ ...
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0answers
32 views

Convolution using Integral Transforms

Returning to the question: Approximation of the convolution operator And new discussion: Convolution using the Laplace integral transform of certain functions $f(t) = e^{-t}$ $g(t) = e^{-(e^{-t})^2}$ ...
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0answers
30 views

Why is the Airy function defined with the pre-factor $(2\pi i)^{-1}$? For normalisation?

The Airy function is a solution to the differential equation \begin{align}y''(x)+x\, y(x)=0.\end{align} Using methods from complex analysis to find solutions to ODEs in terms of contour integrals, it ...
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2answers
37 views

$\int_0^{\infty} a \sqrt{a^2-b^2} \exp \left[ -\frac{a}{c} \right] da= cb^2K_2 \left[ \frac{b}{c} \right]$

How to see that $$\int_0^{\infty} \sqrt{a^2+b^2} \exp \left[ -\frac{a^2+b^2}{c} \right] da= cb^2K_2 \left[ \frac{b}{c} \right],$$ where $$ K_\alpha(x) = \int_0^\infty e^{-x\cosh t} \cosh \alpha t \,dt ...
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2answers
38 views

Equations via Lambert's W function

I'm studying Lambert's W function and I came across the equation $2^x = 2x$. Upon inspection it is easy to see that $x = 1$ and $x = 2$ are the real solutions to the equation. Solving for Lambert's W ...
4
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2answers
87 views

Asymptotic expansion of integral involving Bessel Function and Logarithm

I would like to obtain the asymptotic expression for $b \rightarrow \infty$ of the following integral $$ \tag 1 I = \int_0^1 dx \ln(x) \frac{x}{x^4+a^2} J_0(bx), $$ where $a$ is a real constant and $...
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2answers
435 views

Derive the recurrence relations

Let $P_{m,n}=P_{m,n}(x,y)$ be a polynomial family. Here is some initial terms $$ P_{0,0}=1, P_{1,0}=2x, P_{0,1}=2y, P_{1,1}=8xy.$$ I know that the polynomials for any $m,n \geq 0$ satisfies the five ...
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2answers
58 views

Integral $\int_{0}^{1} \frac{x e^{-cx}}{\sqrt{1-x^2}}\,dx$ (possibly using modified Bessel function)?

Given $c>0$, can the integral $$ \int_{0}^{1} \frac{x e^{-cx}}{\sqrt{1-x^2}}\,dx $$ be expressed in closed form where special functions are allowed? (I can evaluate it numerically for any given ...
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2answers
28 views

bound for the ratio of Gamma functions

Let $x \in R$, $N$ is a natural number. How to bound from above $$ \frac{\Gamma(1-1/x)}{\Gamma(N+1-1/x)} $$
1
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1answer
53 views

History of Gamma and Beta functions

I'm looking for a book on the history of gamma $\Gamma$ and beta $B$ functions! thank you in advance.
1
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1answer
34 views

A lower bound for the cosine integral

I am reading Devroye's paper 2001Simulating Perpetuities. On P103, he mentioned a lower bound for cosine integral, i.e., $$\int_0^t\frac{1-\cos s}{s}ds \geq max(0,\gamma+\log t),$$ where $\gamma$ is 0....
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1answer
37 views

Domain of $\arccos(x)$

Is there an explanation of why the domain of $\arccos(x)$ is $[-1, 1]$?
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1answer
40 views

Is there a way of solving $I_x^{-1}(a_1,b_1)w_1+I_x^{-1}(a_2,b_2)w_2=k$?

Is there any way to rearrange for $x$ in the following formula: $$I_x^{-1}(a_1,b_1)w_1+I_x^{-1}(a_2,b_2)w_2=k$$ $I_x^{-1}(a,b)$ is the inverse incomplete beta function and $1\leq a_1,b_1,a_2,b_2$; $0\...
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0answers
42 views

Big-theta asymptotic limit of expression involving incomplete beta function

I'm having trouble finding the Big-Theta asymptotic limit of expression involving incomplete beta function. $S(k,k)=k\big(\frac{1}{\delta_1}-\frac{1}{\delta_2}\big)+2k^2\dbinom{2k}{k}\bigg[\frac{I_x(k,...
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0answers
27 views

Generalised hypergeometric function $_4F_3$ at unit argument

Is there a way to simplify the following generalised hypergeometric function evaluated at unit argument $$ \, _4F_3\left(-n,3+n,\frac{2}{3},\frac{5}{3};2,-\frac{1}{3}-n,\frac{8}{3}+n;1\right) $$ with $...
1
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1answer
47 views

Inverse of $C\log x + 1/x$ for $x \in (0,\frac 1 C)$.

Let $C > 0$. The function $$\psi_C : (0, \frac 1 C) \to (C(1 + \log(1/C), \infty), x\mapsto C\log x + \frac 1 x,$$ is continuously differentiable and strictly descreasing, since $\psi'_C(x) = \frac ...
3
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2answers
63 views

How do you prove that $\ln(x) = \int_0^\infty \frac{e^{-t}-e^{-xt}}{t}$?

I got the following result using the technique "Integral Milking": $$\ln(x) = \int_0^\infty \frac{e^{-t}-e^{-xt}}{t} dt= \lim_{n\to0}\left(\operatorname{Ei}(-xn)-\operatorname{Ei}(-n)\right)$...
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144 views

Special functions for understanding elemetary integrals

In this earlier question I wrote: Sir Harold Jeffreys wrote: $\dagger$ Consider the integrals $$ I_n = \int_{-1}^1 (1-x^2)^n \cos\alpha x \,dx. $$ Two integrations by parts give the recurrence ...
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0answers
30 views

Which is the function I am looking for?

https://i.stack.imgur.com/P8G7j.jpg Click on the link above to see picture. I have identified the interesting function: (1) f(x)=signx((exp1((absx-1)exp(absx-1)))+1) This function has the ...
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0answers
46 views

Show that $\Gamma(a -S) = \frac{\Gamma(1-a+S)\Gamma(a)}{(-1)^S\Gamma(a-S)}$

If $S$ is an integer and $a$ is a fractional, then $\Gamma(a -S) = \frac{\Gamma(1-a+S)\Gamma(a)}{(-1)^S\Gamma(a-S)}$. I think that it can be used $\Gamma(x)\Gamma(1-x) = \Gamma(a-S)\Gamma(1-a+S)=(a-...
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BBP-Formula expressed as a $_{p}F_{q}$-hypergeometric Function

The Bailey-Borwein-Plouffe digit spigot series for $\pi$... $$\pi = \sum_{k = 0}^\infty \left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} \right) \...
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3answers
61 views

Show that $\int_{0}^{\pi/2} (\frac{1}{\sin^3{\theta}} - \frac{1}{\sin^2{\theta}})^{1/4} \cos{\theta} d\theta = \frac{(\Gamma(1/4))^2}{2\sqrt{\pi}}$

Well, I have shown that $B(n, n+1) = \frac{(\Gamma(n))^2}{2\sqrt{2n}}$ From there I could deduce that $B(1/4, 5/4) = \frac{(\Gamma(1/4))^2}{2\sqrt{\pi}}$, then $n=1/4$. I also know that $B(x, y) = \...
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0answers
33 views

A generalized Euler transformation

The Gaussian hyper-geometric function obeys a number of functional identities which are useful in order to speed up the convergence of the series. A glimpse of the Wikipedia page https://en.wikipedia....
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2answers
27 views

problem about the $\mathscr{E}_1(z)$

we know that: $$ \mathscr{E}_1(z)=\int_z^{+\infty}\frac{e^{-t}}{t}\mathrm{d}t,\quad |\arg(z)|<\pi $$ my question is that : why the $|\mathrm{arg}(z)|<\pi$
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0answers
31 views

How to show that the beta funcction is well defined?

The Euler Beta function is defined by $$B(x,y):=\int_0^1t^{x-1}(1-t)^{y-1}dt.$$ To show that B is well defined, i do the follwing steps, so please tell me if they are true or not ! if $0\leq t\...
1
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1answer
158 views

Definite integral involving hypergeometric functions

I would like to know if the following definite integral of the product of elementary hypergeometric functions is known in closed form $$ \int_0^1\,_2F_1\left(-n,1+n;1;x^3\right) \left[3 \, _2F_1\left(...
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0answers
34 views

Special function expression for the (truncated) moment of the generalized Gamma distribution

The, kind of, truncated moment of the generalized Gamma distribution for positive $a,d$ and $p$ is $$\int_0^\infty (x-k)_+^a x^{d-1}e^{-x^p}dx=\frac1p\int_0^\infty (y^{\frac1p}-k)_+^a\,y^{\frac dp-1}e^...
3
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1answer
78 views

A parameterized log-sine integral equating to square of arctan

I have a really convoluted proof of the following: $$ (1) \quad \quad - \int_0^\pi \frac{\sin(2 y) \log(\sin (y/2)) } {r + 1/r + 2 \cos{(2 y)} } dy = \big(\arctan(\sqrt{r})\big)^2 $$ I proved it for ...
0
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3answers
82 views

Integrate $I=\int_0^{\infty} x^n \, e^{ax+\frac{b}{x}} \, \cos(cx) \, dx$?

Is there an expression in terms of some special functions (or a closed form) of the following integral $$I_n(a,b,c)=\int_0^{\infty} x^n \, e^{ax+\frac{b}{x}} \, \cos(cx) \, dx,$$ $n:$ integer, $a\in\...
2
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2answers
66 views

$\sum_{n=0}^\infty\frac{H_n(x)H_n(y)t^n}{2^nn!}$=$\frac{\exp\left[\frac{2xyt-(x^2+y^2)t^2}{1-t^2}\right]}{\sqrt{1-t^2}}$

I am told to prove that : $$\sum_{n=0}^\infty\frac{H_n(x)H_n(y)t^n}{2^nn!} = \frac{\exp\left[\frac{2xyt-(x^2+y^2)t^2}{1-t^2}\right]}{\sqrt{1-t^2}}$$ where $H_n(x)$ is Hermite polynomial.I am ...
0
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0answers
25 views

Confluent hypergeometric function for integer values in a closed form

I have an issue about finding expression involving finite summation for $_1F_1(mn,n,\delta x)$ where $m$ and $n$ are integer and $m,n>1$ for my case. I found an expression in [T.o.I. 3.383.$1^{11}$]...
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1answer
34 views

Express in terms of Euler integrals [closed]

Express in terms of Euler integrals: $$ \int_{0}^{+\infty}\frac{x^{m-1}}{(1+x)^n} dx $$
0
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1answer
38 views

Show that $P_n(x) ={}_2F_1\left(-n,n+1;1;\frac{1-x}{2}\right)$.

I am told that $$P_n(x) ={}_2F_1\left(-n,n+1;1;\tfrac{1-x}{2}\right),$$ where $P_n(x)$ is Legendre polynomial and ${}_2F_1\left(a,b;c;z\right)$ is hypergeometric function. I am just wondering how to ...
7
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1answer
228 views

Inverse of $f(x)=x^n(1-x)^k$

I am trying to find an inverse of a function \begin{align} f(x)=x^n(1-x)^k, x \in (0,1) \end{align} where $n$ and $k$ are some positive integers. I know that his function doesn't have a 'pure' ...
0
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0answers
34 views

An identity of Hermite polynomials

Given that $$H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$$ Show that $\left|H_n(ix)\right|\geq\left|H_n(x)\right|.$ I wrote that $$\left|H_n(ix)\right|=\left|e^{-x^2}\frac{d^n}{dx^n}e^{x^2}\right|,\...
0
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1answer
36 views

Integral over triple Legendre polynomials involving derivatives

I know the integral over the triple product of Legendre polynomials (see Legendre Polynomials Triple Product), which reads \begin{align} \int_{-1}^{1} P_k(x)\,P_l(x)\, P_m(x) \;\mathrm{d}x = 2 \begin{...
0
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0answers
21 views

How do you find the Inverse of Incomplete Elliptic İntegral of Second Kind when modulus is large

So I tried to take the inverse of EllipticE when k modulus is large, in Mathematica, but the solution gives wrong answer. ...
2
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1answer
65 views

Summation and integral representations of special functions

I have a slightly strange (and possibly quite vague) question that I'm keen to hear people's thoughts on. In my recent work, I have been coming across various infinite series and integrals that cannot ...
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0answers
28 views

Solving equations containing sum of inverse of Lambert W function

Lambert W function, W(z), is defined as the inverse relation of the function $f(w)=we^w$, i.e. if $we^w = z$ then $W(z) = w$. This function is implemented in several software libraries. If I wish to ...
4
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0answers
125 views

Applying the Lambert W function for a new equation

I am a physical chemist, and we are trying to model a simple process. We ended up arriving at the following equation. \begin{align} ax^{-7/2}e^{-bx(ln(cx)-1)}=1 \end{align} We'd like to solve for $...
1
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2answers
30 views

Proof of the expression of upper incomplete gamma function as an finite summation.

I am trying the proof the one of the finite summation expression of the incomplete upper Gamma function $$ \Gamma(m,x) = \int_{x}^\infty v^{m-1}e^{-v}d$$ So, using the equality (also given in Tab. of. ...
0
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1answer
39 views

How to solve equations involving the sum of special exponential functions?

I am studying equations of the form $f_1 e^{g_1z}+f_2e^{g_2z}=f_3e^{g_3z}$ where the $f$'s and $g$'s are functions of the form $ax+by$, with $a, b \in \mathbb{R}$. I'm interested in things like $...
0
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0answers
35 views

Triple product identity for the Gamma function

The well-known reflection formula for the Gamma function relates its values at $z$ and $1-z$ for any $z\in\mathbb{C}$, and has the explicit form \begin{equation} \Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin{\...
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0answers
12 views

Show that $Un (x)=\frac {(-2)^n (n+1)!}{(2n+1)!}(1-x^2)^\frac {-1}{2}\frac {d^n}{dx^n}(1-x^2)^{({n}+{\frac {1}{2})}}$

Show that $Un (x)=\frac {(-2)^n (n+1)!}{(2n+1)!}(1-x^2)^\frac {-1}{2}\frac {d^n}{dx^n}(1-x^2)^{({n}+{\frac {1}{2})}}$ I want to show that Un where $Un=\binom {n+1}{1}x^n-\binom {n+1}{3}x^{n-2}(1-x^2)+\...
0
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0answers
58 views

Summation of a quotient with a square root

Is anyone aware of an expression for the following series, presumably involving a special function? $$ \sum_{n=-\infty}^{\infty}\frac{e^{inx}}{\sqrt{k^{2}-(n+\alpha)^{2}}(n+\beta)} $$ Mathematica won'...
3
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0answers
78 views

Evaluating $\int_0^\pi\frac{e^{-\sin x}}{e^{-\sin x}+e\sin x}dx$

I think I thought of this integral a little over a year ago and I just haven't been able to do it. I really want to and everytime I sit down with it I learn a little bit more (or atleast I think I do) ...

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