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Questions tagged [special-functions]

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

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Integral of a multivalued function

I am trying to calculate an integral of a multivalued function, which has the form: $\begin{align} I=&\int_0^{\infty} d\tau \ \left( \frac{1}{v_c \tau+\alpha} \right)^{\frac{1}{4}(K_c+1/K_c)} \...
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Integral of Bessel function multiplied with sine $\int_c^\infty J_0(bx) \sin(ax) dx$. [on hold]

I want to do $\int_c^\infty J_0(bx) \sin(ax)$ I know the answer where the limit of integral start from zero. Any help would be much appreciated. Thanks in advance.
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59 views
+50

Integrals involving powers and beta function

I have the three following integrals, very similar the one to the others, $$I_1^{(p)}(N)\equiv\frac{1}{2^{N+p}}\int_0^1(1+t)^{N-1}(1-t)^pB\left(\frac{1}{t+1};N+p+1,N\right)\text{d}t$$ $$I_2^{(p)}(...
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7 views

What differential equations describe absolute squares of Hankel-like functions of real numbers?

The linearly independent solutions of Bessel equation can be combined into two complex functions, which would represent running radial cylindrical waves. These are the Hankel functions $H_\alpha^{(1)}$...
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1answer
17 views

Orthonormality of Hermite function

I was wondering if someone could tell me when the following relation holds? where $H_{n}(x)$ are Hermite polynomials and $\delta(x-x')$ is Dirac delta function: $$ \sum_{n=0}^\infty \frac{1}{\sqrt{\pi}...
2
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1answer
94 views

Double integrals involving incomplete beta function

I am trying to solve without success the following double integral $$I_1^{(p)}(N)\equiv\frac{1}{2^p}\int_0^1\text{d}x\int_0^1\text{d}y(1+y-x)^{N+p}(1+x-y)^{N-2}B\left(\frac{1}{1+y-x};N,p+1\right)\...
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74 views

Is there an identity between the Clausen function $\rm{Cl}_8\left(\frac\pi3\right)$ and $\sum_{n=1}^\infty \frac{1}{n^9\,\binom {2n}n}$?

Given the log sine integral, $$\rm{Ls}_m\Big(\frac{\pi}3\Big) = \int_0^{\pi/3}\Big(\ln\big(2\sin\tfrac{\theta}{2}\big)\Big)^{m-1}\,d\theta$$ we have in this post, $$\begin{aligned} \frac\pi{2!}\,\...
2
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1answer
71 views

Integral $\int_0^\infty\frac{\exp(i\alpha\cos u)-J_0(\alpha)}{1+\beta u}\mathrm{d}u$

I was studying the motion of a particle in a certain magnetic field and one of the quantities that arose was given by the titular integral $$ F(\alpha,\beta)=\int_0^\infty\frac{\exp(i\alpha\cos u)-J_0(...
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2answers
57 views

How to integrate $\sqrt{\arctan(x)}$ [closed]

How to do $$\int\sqrt{\arctan(x)}\, \mathrm dx \:??? $$ Is there any other special function defined like this?
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1answer
39 views

Asymptotic expansion of incomplete beta function

I would like to write down an asymptotic expansion in the $N\to\infty$ limit of the following incomplete beta function $$B\left(\frac{N}{N+1};N,p+1\right)=\int_0^{\frac{N}{N+1}}x^{N-1}(1-x)^p\,\text{...
1
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1answer
97 views

Integral $\int_0^1 \sqrt{\frac{x^2-2}{x^2-1}}\, dx=\frac{\pi\sqrt{2\pi}}{\Gamma^2(1/4)}+\frac{\Gamma^2(1/4)}{4\sqrt{2\pi}}$

I have been trying to find the arc-length of $\sin^{-1}(x)$ over $[0,1]$. Of course, it is given by the integral $$J=\int_0^1\sqrt{1+\frac1{1-x^2}}\ dx=\int_0^1 \sqrt{\frac{2-x^2}{1-x^2}}\, dx$$ To ...
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7 views

Partial derivatives of m=l spherical harmonics

The spherical harmonics, $Y^m_n(\theta, \varphi)$, are only defined when $m$ $\in$ $[n,n-1,...,-n+1,-n]$. However, the derivative relation with respect to $\theta$ requires $Y^{m+1}_n(\theta, \varphi)$...
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0answers
37 views

Names for specific functions

Well known functions are the power function $x^k$ and the exponential function $a^x$. What name could be given to the function $$f(x) \, = \, \frac{x^k}{k^x}?$$ The standard logistic function is ...
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103 views

Sum involving incomplete beta functions

I am interested in the evaluation (A), or at least an asymptotic expansion for large $N$ (B), of the following finite sum \begin{equation}\begin{split} S^{(p)}(N)&\equiv2N\sum_{k=1}^{\left\...
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0answers
15 views

Relation for two the Fox-H function with positive and negative argument

Is there the relation between Fox-H function with positive argument and Fox-H function with negative argument? My question is attached in the image.
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0answers
12 views

Relation between two Fox-H function with positive and negative argument

Is there the relation between Fox-H function with positive argument and Fox-H function with negative argument? My question is attached in the image.
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1answer
39 views

Closed form of $\int_0^1 (u-1/2)^n(1-x\, u)^a(1-y\,u)^b\,du$

Working on a problem related with Apple function, I arrive to an expresion with the following integral representation $$\int_0^1 (u-1/2)^n(1-x\, u)^a(1-y\,u)^b\,du$$ with $a,b,x,y\in\mathbb{R}$ and $...
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2answers
109 views

Evaluate $\sum\limits_{n\geq1}\frac{(-1)^n}{3^n(2n+1)}\sum\limits_{k=1}^{n}\frac{(-1)^k}{k}{n\choose k}(x^k-1)$

In my answer here, I reduce the problem of evaluating $$J=\int_0^{\pi/6}\frac{x\cos x}{1+2\cos x}dx$$ to the evaluation of $S(8-4\sqrt3)$, were $$S(q)=\sum_{n\geq1}\frac{(-1)^n}{3^n(2n+1)}\sum_{k=1}^...
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Legendre Polynomial of Second Kind-Neumann's Formula

In textbook Mathews&Walkers problem 7.6 Starting from \begin{equation*} Q_n(z)=\frac{1}{2} P_n(z)\ln\left( \frac{z+1}{z-1}\right)+f_{n-1}(z) \end{equation*} we can derive Neumann's Formula \begin{...
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1answer
40 views

Series with gamma functions

I would like to understand how can I write down the expression for the following series: $$S_0=\sum_{k=2}^{\infty}(-1)^kA^{k}\frac{\Gamma(k-3/2)}{\Gamma(k+1)}.$$ I have seen related topics on this ...
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1answer
41 views

An Equation Involving the Trigamma Function

Let $N$ be a positive integer, and consider the equation \begin{equation} \frac{1}{2} \sum_{n=1}^{N} \psi^{(1)} \left( \frac{x+1-n}{2} \right) = \frac{N}{x}, \end{equation} in the real unkown $x > ...
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1answer
20 views

A System Involving the Trigamma Function

Given $K \geq 2$ real numbers $a_1, \dots, a_K$, with $a_k > 0$ for $k=1,\dots,K$, consider the system of equations \begin{equation} (a_k - x_k) \psi^{(1)}(x_k) = \psi^{(1)} \left(\sum_{k=1}^{K}...
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1answer
59 views

Sum of two Fox H-functions

I want to add up the following two Fox H-functions $$ H_{1,2}^{\,1,1} \!\left[ -\lambda^2 \left|x\right|^{2\alpha^\prime} \left| \begin{matrix} ( 0 , 1 ) \\ ( 0 , 1 ) & ( 0 , 2\alpha^\prime ) \...
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0answers
36 views

Use Meijer-G function to represent the products of two Bessel functions

Here I define two modified Bessel functions of the second type: $K_{\mu }\sqrt{az}$, $K_{v} \sqrt{\frac {b}{z}}$. I am struggling to represent the products of these two modified bessel functions by ...
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Trigonometric integral related to Gieseking's constant

This question at MathOverflow https://mathoverflow.net/questions/302982/how-to-prove-the-identity-l2-frac-cdot3-frac215-sum-limits-k-1-inf conjectures certain relation between fast converging ...
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31 views

Heaviside multivariable integral on a finite domain

I have an integral of the form $$I=\int_0^{\ell_p}dp_1\int_0^{\ell_p}dp_2\theta\left(-\lambda+\cos\left(\frac{2\pi p_1}{\ell_p}\right)+\cos\left(\frac{2\pi p_2}{B\ell_p}\right)\right),$$ Where $B,\...
4
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1answer
46 views

Beta function in Philip J. Davis׳ Essay

This question is about equation number (4) in Philip J. Davis’ Essay titled "LEONHARD EULER'S INTEGRAL: A HISTORICAL PROFILE OF THE GAMMA FUNCTION". In there it is stated by the author "Euler ...
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1answer
19 views

Show that second linearly independent solution to legendre ODE is $Q_n(x)=P_n(x)\int^x \frac{dx}{(1-x^2)[P_n(x)]^2}$

$$Q_n(x)=P_n(x)\displaystyle\int^x \dfrac{dx}{(1-x^2)[P_n(x)]^2}$$ The form looks like Green's function, or general solution after the variation of parameters method. I couldnot figure it out. I ...
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26 views

Patterns for the polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ and Nielsen polylogarithm $S_{n,p}\big(\tfrac12\big)$?

The polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ has closed-forms known for $m=1,2,3$. It seems this triad pattern extends to the Nielsen generalized logarithm $S_{n,p}\,\big(\tfrac12\big)$ for $n=0,1,...
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2answers
202 views

Closed-forms for the integral $\int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$?

(This is related to this question.) Define the integral, $$I_n = \int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$$ with polylogarithm $\rm{Li}_n(x)$. Given the Nielsen generalized polylogarithm $S_{n,p}(z)$, $$...
2
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Integral $\int_0^x \ln^{n}\left(2\sin\frac{t}2\right)\,dt$

Evaluate $$\mathcal{L}_n(x)=\int_0^x \ln^n\left(2\sin\frac{t}2\right)\,dt\qquad n\in\Bbb N_0, x\in[0,\pi/2]$$ In this answer to a question of mine, @TitoPiezasIII claims that $$\frac{(-1)^n}{n!}\...
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160 views

More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$

In this post, the OP asks about the integral, $$I = \int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$$ I. User DavidH gave a beautiful (albeit long)...
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1answer
38 views

Chebyshev polynomial property

I want to prove inequality (5.13) but I have a problem with (5.16). I have: $$ \sin(n\theta) = \sin\theta \cos(n-1)\theta + \sin(n-1)\theta \cos\theta = $$ $$ = \sin\theta \cos(n-1)\theta + \cos\...
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1answer
570 views

Does anyone recognize the function from this picture?

I was playing with the exterior algebra, and stumbled on this interesting function from $\Bbb N^2 \to \Bbb N$, which I'll call $f(x,y)$. This is plotted from $1 \leq x,y \leq 100$: In this picture, ...
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0answers
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Help with an identity involving PolyLog and Logs.

I am plotting the following expression on Mathematica $$ \Im\left[-96 \text{Li}_3\left(e^{-i t}\right)-48 t^2 \log \left(1-e^{-i t}\right)\right]-96 \Re\left[t \text{Li}_2\left(e^{-i t}\...
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2answers
147 views

Integral $\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$

Prove that $$\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$$ I was given this integral in my post Request for crazy integrals. I have never seen an integral like this before ...
2
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1answer
107 views

Integral: $\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$

I am trying to evaluate $$P=\frac\pi2\sum_{n\geq1}\frac{{2n\choose n}}{4^n n^2}$$ I used the beta function to show that $$P=\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$$ IBP: $$P=\sin^{-1}(x)\...
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0answers
29 views

Modified Airy functions

The question is quite formal. I recall the definition of Airy function $$Ai(\tau^{2/3}\zeta)=\frac{\tau^{1/3}}{2\pi}\int e^{i(\sigma^3/3+\sigma\zeta)}d\sigma,\quad Ai'(\tau^{2/3}\zeta)=\frac{i\tau^{1/...
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1answer
51 views

Is Lerch's transcendent a multi-valued function?

Lerch’s Transcendent is defined by $${\Phi\left(z,s,a\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$$ when $|z|<1$ or $\Re s>1,|z|=1$. If $s$ is not an integer then $|\mathrm{ph}(a)|<\...
1
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1answer
62 views

Transformations relating 3F2 at z with 3F2's at 1/z

I am searching for some transformations for a 3F2 hypergeometric function which send the argument z to 1/z. I am aware of the one given in NIST book (p. 410, Formula 16.8.8) in the special case q=2 (...
15
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2answers
409 views

Integral $T_n=\int_{0}^{\pi/2}x^{n}\ln(1+\tan x)\,dx$

For $n\in\Bbb N_0$, evaluate in closed form $$T_n=\int_{0}^{\pi/2}x^{n}\ln(1+\tan x)\,dx$$ After seeing @mrtaurho's answer to this question, I realized that it would be possible to generalize his ...
0
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1answer
48 views

Finding the $m^\text{th}$ term of an expression?

How to find the $m^\text{th}$ term for the following expression: $$ \left.\frac{\partial^m}{\partial s^m}e^{a s^2}\right|_{s=0}$$ Is there any analytical approach? I computed first few terms ...
2
votes
1answer
43 views

Connecting integral involve integer and half-integer Bessel function

Suppose I have that: $$ \int_0^\pi xJ_0(\alpha x)g(x) dx \geq 0 \ \ \ \ (*) $$ then is there anything I can say about: $$ \int_0^\pi xJ_{\frac{-1}{2}}(\alpha x)g(x) dx \ \ (\geq 0?) \ \ \ (**)$$ ...
1
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0answers
39 views

Mittag-Leffler function and Fox-Wright function

I find the following identity in many special functions books without proof. This identity is called the Laplace transform of the Mittag-Leffler function with three parameters. The result is in the ...
0
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1answer
31 views

What are the poles and zeros of the Euler Beta function?

For what pairs of complex values $(x,y) \in \mathbb{C} \times \mathbb{C}$ does the Euler Beta function $B(x,y)$ equal zero? For what pairs of complex values $(x,y) \in \mathbb{C} \times \mathbb{C}$ ...
2
votes
2answers
112 views

Computing the integral $x^2\textrm{sech}^2(x)$

I'm trying to compute the integral $$\int_{0}^{\infty}dx \, x^{2}\operatorname{sech}^{2}(x)=\frac{\pi^{2}}{12}.$$ Manually, one obtains, quite naively, $$\int dx \, x^{2}\operatorname{sech}^{2}(x)=\...
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0answers
48 views

Evaluating the integral x/(e^(x)+1) [duplicate]

I'm trying to evaluate the following integral $$I=\int_{0}^{\infty}dx\frac{x}{e^{x}+1}=\frac{\pi^{2}}{12},$$ where the result comes from Mathematica. Manually, one has $$\int dx\frac{x}{e^{x}+1}=\...
3
votes
1answer
91 views

Prove an transformation formula for Gauss hypergeometric function $_2F_1(a,b;c;z)$

In " Special functions: an introduction to classical functions of mathematical physics" by Nico M. Temme, at page 113 is reported this formula: $$_2F_1(a,b;c;z)=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\...
2
votes
1answer
34 views

Equation involving difference of beta CDFs

Consider the expression $$I_{p}(\alpha,\beta+1) - I_{p}(\alpha+1,\beta) = c$$ where $I_p(a,b)$ is the regularized incomplete beta function. Question: Given $\alpha,\beta,$ and $c>0$, what is $p$? ...
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0answers
40 views

Looking for an unbounded, monotonic, non-symmetric s-shaped function

I am hoping to find a function that corresponds to an s-shaped curve that satisfies the following properties: 1. unbounded 2. it is not symmetric about its inflection point 3. its derivative at the ...