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).

Filter by
Sorted by
Tagged with
0 votes
0 answers
16 views

From incomplete beta function sum $\frac1{(\text B(a,b)c)^2}\sum_{k=0}^\infty\frac{\text B_y(2a+r+k,b)(1-b)_k}{(a+k+r)k!}$ to hypergeometric function.

The goal is to integrate Inverse Beta Regularized $\text I^{-1}_{z}(a,b)$ to a constant power with respect to $z$ twice for a future identity. Notice the Incomplete Beta function $\text B_z(a,b)$ and ...
user avatar
  • 5,229
2 votes
1 answer
48 views

Asymptotic behavior of the integral $\int_{0}^{\infty} e^{-\rho \cosh(R)}\cosh(Rq)dR$

Which method can I use to study the asymptotic behavior as $\rho \to \infty$ of the integral for $q \geq 0$? $$\int_{0}^{\infty} e^{-\rho \cosh(R)}\cosh(Rq)dR$$ I wish to study this behavior to ...
user avatar
0 votes
0 answers
40 views

Integrating $\int_{1}^{\alpha} x^n(x^2-1)^{q-\frac{1}{2}}dx$

Let $\alpha>1$, $n \in \mathbb{N}$ and $q\geq0$. Which methods is possible to use to solve this integral? $$\int_{1}^{\alpha} x^n(x^2-1)^{q-\frac{1}{2}}dx$$ I tried using the computer for especific ...
user avatar
0 votes
0 answers
42 views

Integral representation of modified Bessel function of second kind

I'm looking for a proof of the following integral representation for the modified Bessel function of the second kind $K_q(\rho)$ for $q \geq 0$ and $\rho>0$ $$K_q(\rho)=\frac{\Gamma(1/2)(\rho/2)^q}{...
user avatar
0 votes
0 answers
53 views

Asymptotic behavior of modified Bessel function help

The modified Bessel function of the first kind is defined as $$I_q(\rho)=\sum_{m=0}^{\infty} \frac{\left(\frac{\rho}{2}\right)^{2m+q}}{m!\Gamma(m+q+1)}$$ where $\rho \in \mathbb{C}\setminus \{0\}$ and ...
user avatar
1 vote
2 answers
95 views
+200

What is the $t$-anti-derivative $\int dt \; e^{i t}\mathrm{Ei}\big(- i [t+x]\big)\; t^{-n}$ for $n = 1,2,3$?

Consider the functions for $n = 1, 2, 3$ $$ f_{n}(t,x) := \frac{e^{i t}\mathrm{Ei}\big(- i [t+x]\big)}{t^{n}} \ , $$ where $\mathrm{Ei}$ is the exponential integral function defined for complex $z \...
user avatar
0 votes
0 answers
21 views

Discontinuity in Legendre Function of the Second Kind

The Legendre function of the first kind, $P_{\nu }(z)$, is usually defined in a way that it has a branch cut along the segment ($-\infty <z\leq -1$] while the Legendre function of the second kind,$...
user avatar
4 votes
0 answers
93 views

Question about the solution to the heat equation in spherical coordinates

I was solving the heat equation in spherical coordinates with standard boundary conditions: temperature held at 0 at the boundary $r=\alpha$. I was able to find all eigenvalues and eigenfunctions. I'm ...
user avatar
  • 284
3 votes
1 answer
63 views

Analytic formula for $f(\alpha):= \int_{-\infty}^\infty\frac{|e^{is}-1|}{|s|^{2\alpha+1}}\,ds$

For any $\alpha \in (0,1)$, define $\displaystyle f(\alpha):= \int_{-\infty}^\infty\frac{|e^{is}-1|}{|s|^{2\alpha+1}}\,ds$. Question. Is there an analytic formula for $f(\alpha)$, say, in terms of ...
user avatar
  • 8,289
8 votes
3 answers
158 views

A gamma summation: $\sum_{n=0}^{\infty} \frac{2}{\Gamma ( a + n) \Gamma ( a - n )} = \frac{2^{2a-2}}{\Gamma ( 2a - 1 )} + \frac{1}{\Gamma^2 (a)}$

Let $a \notin \mathbb{Z}$ and $a \neq \frac{1}{2}$. Prove that $$\sum_{n=0}^{\infty} \frac{2}{\Gamma \left ( a + n \right ) \Gamma \left ( a - n \right )} = \frac{2^{2a-2}}{\Gamma \left ( 2a - 1 \...
user avatar
  • 6,540
1 vote
0 answers
46 views

Lagrange inversion theorem of $x^r(x+k)$ to generalize the W Lambert function

Motivation: $2$ branches of Lambert $\text W_k(z)$ is a limit of the inverse of $x^n(x+c)$ which is expressible in terms of FoxH in Mathematica. $\text W_0(x)=\text W(x):$ $$-\lim_{a\to0}e^{\frac{(-x)^...
user avatar
  • 5,229
-1 votes
1 answer
43 views

beta functions that don't reduce to known values

I am trying to convert $B(13/3, 11/3)$ into a Gamma function. I was hoping to reduce to $\Gamma(1/3 )$ for which there is a fixed value, but of course one reduces to $(6930/243) \Gamma(2/3)$ and the ...
user avatar
  • 9
0 votes
0 answers
36 views

Inverting the sum of two logarithms?

It is well known how to invert the following relation to get $y$ in terms of $x$ $$x=a\log(y-b),\quad\to\quad y=b+\exp(x/a)$$ Is there anything that can be said about inverting the relation $$x=a\log(...
user avatar
  • 731
0 votes
1 answer
35 views

For what reason Bessel functions of the first kind can be differenciated in relation to the variable $q$?

For that reason the modified bessel function $I_q(\rho)$ defined as $$ I_{q}(\rho)=\sum_{m=0}^{\infty}\frac{(\rho/2)^{2m+q}}{m!\Gamma(m+q+1)},$$ where in the above $\rho$ is fixed, can be ...
user avatar
0 votes
1 answer
21 views

A Regularized Beta function limit: $\lim_{a\to0}\frac{1-\text I_\frac zb(a,b)}a$

The goal is to “generalize” the Exponential Integral $\text{Ei}(x)$ using the Regularized Beta function $\text I_z(a,b)$: $$f(b,z)=\lim_{a\to0}\frac{1-\text I_\frac zb(a,b)}a$$ Some clues include: $$\...
user avatar
  • 5,229
2 votes
2 answers
45 views

Simplifying a summation involving factorials

How does one show this? $$ \exp(-x) \sum_{k=0}^\infty x^k \frac{(k+m)!}{(k!)^2} = L_m(-x) m!, $$ where $m$ is a positive integer, and $L_{m}(x)$ is the $m$th order Laguerre polynomial.
user avatar
  • 436
4 votes
1 answer
91 views

Does the Incomplete Beta function have forms of Elliptic E besides $\frac14 \text B_{\sin^2(2x)}\left(\frac12,\frac34\right)=\text E(x,2)$?

Goal: To find more special cases of the Incomplete Beta function $\text B_z(a,b)$ in terms of Elliptic $\text E(x,k)$ using Mathematica notation: The goal is to find values of: $$\text B_z(a,b)=\int z^...
user avatar
  • 5,229
2 votes
1 answer
64 views

Fourier transform of $\frac{(x-iw)^\alpha}{(x+iw)^\alpha}$

Let $w>0$ and $\alpha>0$. I want to compute the Fourier transform of $\frac{(x-iw)^\alpha}{(x+iw)^\alpha}$ in the distribution sense, i.e., evaluate $$\int_{-\infty}^\infty \frac{(x-iw)^\alpha}{(...
user avatar
  • 1,675
0 votes
1 answer
42 views

What are the elementary properties of dirac-delta function from which every other properties of it could be deduced?

I am studying dirac-delta function first time in my undergraduate course and different books have defined this function in different ways which when graphed together contradicts each other. I want to ...
user avatar
3 votes
0 answers
117 views

How can we prove a closed form for $\frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right)+\text{Li}_2\left(2+\sqrt{3}\right)$?

I have been working on a problem in number theory that I have reduced to the problem of showing that the two-term linear combination $$ \frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right) + \...
user avatar
0 votes
0 answers
77 views

Upper bound for Meijer-G function

What I need is a monotonically decreasing function that forms an upper bound for the following function: \begin{equation} - G_{0,6}^{4, 0}\biggl({-\atop -\frac{1}{2},\frac{1}{2},\frac{5}{6},\frac{7}{6}...
user avatar
  • 1
2 votes
1 answer
44 views

Mittag-Leffler function recurrence relation

The general Mittag-Leffler function $$E_{a,b}(z)=\sum_{h=0}^{\infty}\frac{z^h}{\Gamma(ha+b)}$$ satifies the recurrence $$E_{a,b}(z)=zE_{a,b+a}(z)+\frac1{\Gamma(b)}.$$ I am having a hard time in ...
user avatar
  • 6,557
1 vote
0 answers
32 views

What is the correct representation of the generalized gamma function?

The NIST Digital Library of Mathematical Functions defines the multivariate gamma function as $$ \Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}\mathrm{etr% }\left(-\mathbf{X}\...
user avatar
  • 554
0 votes
0 answers
17 views

Proof $x^aJv(bx^c)$ solves the equation $ y'' - (\frac{2a-1}{x}) y'+ (b^2c^2x^{2c-2} + \frac{a^2-v^2c^2}{x^2}y) =0$

I have been trying to plug the function into the equation and see what comes out, but I have not been able to make any progress. Is this the right approach? You don't have to show me all the work, but ...
user avatar
  • 284
4 votes
0 answers
125 views

Are there applications for the inverse of the arc length of $ax^n$ and $a^x$? “Closed forms” found.

Based on: How to straighten a parabola? and Arc length of $x^n$ found using Hypergeometric function and series. Alternate representations and solution verification needed. Use: $$\text{ArcLength}(...
user avatar
  • 5,229
1 vote
1 answer
46 views

special function that is between arcsinh(x) and arctan(x)? [closed]

I am looking for a function between arcsinh(x) and arctan(x) for x>0. The function should be calculated using +, -, *, /, elementary functions, or well know special functions. But no numerical ...
user avatar
1 vote
0 answers
19 views

Extension of reflection formula for polygamma function

So I've seen the reflection formula for polygamma functions: $\Psi(z) - \Psi(1-z)=-\pi \cot{\pi z}$. Is there an extension for arguments that sum not to $1$ but to some other (not necessarily integer) ...
user avatar
1 vote
3 answers
81 views

Can you suggest a software to solve this equation?

Can you please suggest a free software or website that would allow me to approximate, numerically, the first $n$ roots of the equation $ 2J_{0}(2\alpha)+2\alpha J'_{0}(2\alpha)=0$ ? I'm trying to find ...
user avatar
  • 284
1 vote
0 answers
22 views

Intersection of modified Bessel functions with different scaling

Let $0< s_1 < s_2$ and $0<Z_1<Z_2$, consider functions $F_j:[0, \infty)\to [0, \infty)$ defined by $$F_j(t) = \frac{1}{Z_j} I_0(2\sqrt{ts_j}),$$ where $j = 1,2$ and $I_0$ is the modified ...
user avatar
1 vote
0 answers
79 views

Evaluation of $\sum_{n=1}^{\infty}\frac{1}{n(n!)}$

WolframAlpha gives the evaluation as $\text{Ei}(1)-\gamma$. It does not offer a step-by-step solution. Where does this come from?
user avatar
5 votes
0 answers
102 views

Does $(1-\cos(x))/x^2$ have or deserve a name, like $sinc$ for $\sin(x)/x$?

i think it's the “dual” of $\mathrm{sinc}(x)$, that is, $$f(x) = \begin{cases} \frac 1 2, & \text{if $x=0$} \\ \frac{1-\cos{x}}{x^2}, & \text{otherwise} \end{cases}$$ they share some ...
user avatar
3 votes
1 answer
36 views

Evaluating an integral with derivatives of Associated Legendre polynomials

I came across the following integral $$\int_{-1}^{+1} (1-x^{2}) \frac{\partial P_{lm}(x)}{\partial x} \frac{\partial P_{km}(x)}{\partial x} dx$$ where $P_{lm}(x)$ is an associated Legendre polynomial, ...
user avatar
  • 361
0 votes
1 answer
20 views

How to find $xy=F(t)$ from $- \operatorname{Ei}{\left(x y e^{i \pi} \right)} = A t^{2} + B t + C$?

I have the integral equation $$- \operatorname{Ei}{\left(x y e^{i \pi} \right)} = A t^{2} + B t + C$$ where $\operatorname{Ei}$ is the exponential integral and $A$, $B$, and $C$ are (finite) arbitrary ...
user avatar
  • 1,316
0 votes
0 answers
17 views

Double sum in terms of generalized Mittag-Leffler functions

I have the series $$S = \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{x^n y^m}{\Gamma(1+\alpha n + \beta m)}\frac{(n+m)!}{n!m!}$$ which originates from a fractional calculus problem. One can see that $S$ ...
user avatar
0 votes
0 answers
14 views

Inverse Laplace transform of $ \frac{1}{s}\frac{1}{1 + (k/s)^a + (l/s)^b}$

I want to express the Inverse Laplace transform (arising from a fractional calculus problem) $$ F(t) = \mathcal{L}^{-1}\Big\{ \frac{1}{s}\frac{1}{1 + (k/s)^a + (l/s)^b}\Big\}(t) $$ in terms of ...
user avatar
1 vote
0 answers
72 views

Integral of $e^{-(Ax^2 + By^2)}$ around the unit sphere

I'm trying to calculate $\mathcal{I} = \int_{S^2} e^{-(Ax^2 + By^2)} dS$ for real constants $A$ and $B$, where $S^2$ is the two-dimensional sphere.. I have external constraints on these constants ...
user avatar
0 votes
0 answers
28 views

An elegant computation of series

I try to compute the following series, $$S(\omega,k)=\sum_{p=0}^{\infty}\frac{(-k)^p(p+k)^2}{(p+k)^2+\omega^2}\frac{1}{p!},$$ where $\omega>0$, $k>0$. To be honest, I have no idea how to compute ...
user avatar
0 votes
0 answers
16 views

Calculate time until accrued compound interest reaches a fixed amount.

Suppose I start with an amount $a_i$ of each of several assets indexed $i\in \mathbb{I}$. The fixed interest rate associated with each asset is $r_i$. After some time $\Delta t$ the amount held of ...
user avatar
0 votes
0 answers
21 views

Derivative of Bessel function of order 1

what is the derivative of the given Bessel function? d/dx(xJ1(x)) , where x=A.z A is constant and z is variable. If one Bessel function is J1(z) and the other is J1(Az), Is it possible to write the ...
user avatar
  • 11
0 votes
0 answers
19 views

Does the Meier-G function have a lower and an upper bound?

The Meijer G-function is given by It serves as a solution to certain differential equations, and has been shown to generalize a variety of elementary and transcendental functions. Is there a lower ...
user avatar
  • 1,316
0 votes
1 answer
45 views

Simplifying a ratio of incomplete beta functions

Can the following be simplified? $$ \frac{\int_0^a t^{x+1} (1-t)^{y} dt}{\int_0^a t^{x} (1-t)^{y} dt} \qquad \big((x,y)\in(0,\infty)^2; a\in(0,1]\big) $$ Note: If it helps to assume that $0<x<y$ ...
user avatar
0 votes
0 answers
24 views

Simplifying a ratio of "two-sided incomplete Beta functions"

Let $0\leq L < R\leq 1$ and $0\leq s \leq T$. Can the following ratio be $\text{simplified?}$ $$ \frac{\int_L^R z^{s+1}(1-z)^{T-s}dz}{\int_L^R z^{s}(1-z)^{T-s}dz} $$ If it helps to assume that $s$ ...
user avatar
0 votes
1 answer
89 views

Strange summation? $\sum _{k=-n}^{n+1} \frac{(-1)^k}{x-k}$

I'm mainly concerned with the bounds of summation here. I've never personally seen such a summation before, but I came across this summation in "Special Functions" by Andrews Askey Roy on ...
user avatar
1 vote
1 answer
38 views

How to solve the following set of integrals by using Hypergeometric functions?

Any recommendations to help me to solve this integral $$ \int_{-\sqrt{x}+y}^{1-y} \left(1-(t+y)^2\right)^{r} \left(1-x(t-y)^2 \right)^{r} dt $$ where $\{r,x,y\} \in \mathbb{R} , \, |y|<1+x, ...
user avatar
  • 55
1 vote
0 answers
70 views

Double sum over Gauss hypergeometric function.

I've been dealing with sums and integrals over hypergeometric functions quite a bit lately, and the latest problem is the following double sum: \begin{equation} F(x,y;\alpha,t)=\sum_{n,m=0}^\infty\...
user avatar
  • 171
0 votes
1 answer
29 views

It is $x(t) = e^{c_1-t}\cdot\theta(c_1-t)$ a solution to $\dot{x} = -|x|$ with $\theta(t)$ the unitary step function?

It is $x(t) = e^{c_1-t}\cdot\theta(c_1-t)$ a solution to $\dot{x} = -|x|$ with $\theta(t)$ the unitary step function? I am trying to understand solutions of finite duration to differential equations. ...
user avatar
  • 789
1 vote
1 answer
70 views

How to "formally" prove that $x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$ solves $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$?

How to "formally" prove that $x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$ solves $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$? (with $\theta(t)$ the standard unitary step function). I have found the ...
user avatar
  • 789
2 votes
0 answers
38 views

Finding a simple differential equation to define an inverse of $\,_2\text F_1(a,b;c;z)$ with respect to $z$ with the Gauss Hypergeometric function.

An “Inverse Gauss Hypergeometric function” with respect to $z$ in terms of a differential equation would define many special case inverse functions. Define: $$\,_2\text F_1(a,b;c;z)=\sum_{n=0}^\infty \...
user avatar
  • 5,229
3 votes
1 answer
59 views

Laplace transforms of products of modified Bessel Functions

I am dealing with integrals of the form $$\int_0^\infty e^{-t}I_0(xt/a)^a\ \mathrm{dt}$$ where $I_0(x)$ is the modified Bessel function of the first kind. Clearly this is just a Laplace Transform $\...
user avatar
1 vote
0 answers
28 views

Solution of equation containing two logarithms

I need to solve the following equation for x: $a\log(\frac{x}{a})-b\log(\frac{a-x}{2b})=x-d$. I know that the solution of $\log(a x+b)=c x+d$ can be determined using the Lambert function https://en....
user avatar
  • 21

1
2 3 4 5
85