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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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+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)}(...
20
votes
2answers
669 views

Other challenging logarithmic integral $\int_0^1 \frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx$

How can we prove that: $$\int_0^1\frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx=\frac{\pi^2}{8}\zeta(3)-\frac{27}{16}\zeta(5) $$
2
votes
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)\...
11
votes
3answers
881 views

Ramanujan Summation

It seems that under the light of Ramanujan Summation the following is plausible: $$1 + {2^{2n - 1}} + {3^{2n - 1}} + \cdots = - \frac{{{B_{2n}}}}{{2n}}(\Re)$$ Alas, I can't really find any ...
4
votes
1answer
161 views

Hunt for exact solutions of second order ordinary differential equations with varying coefficients.

Let $a,a_1,a_2,b \in {\mathbb R}$. Being inspired by the answer to Solve $y''(x)=[a(x^2-1)^2+b]y(x)$ we found solutions of the following second order ODE : \begin{equation} \frac{d^2 y(x)}{d ...
4
votes
2answers
137 views

Hypergeometric series for $\mathrm{Cl}_2(\pi/3)$

I am trying to find a hypergeometric series for $\mathrm{Cl}_2(\pi/3)$, where $$\mathrm{Cl}_2(x)=-\int_0^x\log\left|2\sin\frac{t}2\right|dt=\sum_{k\geq1}\frac{\sin kx}{k^2}$$ Is the Clausen ...
2
votes
2answers
1k views

Euler Gamma function $\Gamma(z)$ on $\mathbb{C}$

I'm working on an exercise about the Gamma Function from Euler. First, $\Gamma (z)= \int_0^\infty e^{-t}t^{z-1}dt$. Now, if we consider the "similar" function $\int_{\frac{1}{n}}^\infty e^{-t}t^{z-1}...
7
votes
0answers
118 views

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 ...
0
votes
0answers
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 $H_\alpha^{(1)}$...
2
votes
1answer
70 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(...
0
votes
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}...
5
votes
0answers
71 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
votes
1answer
69 views

Definition of Poincaré rank

Dealing with the confluent Heun equation, something unimportant to me at the beginning got me curious lately: the Poincaré rank of an irregular singularity. In particular, that one in the confluent ...
4
votes
0answers
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\...
2
votes
2answers
289 views

Hurwitz zeta function

I am using the below function to compute the Hurwitz zeta function from Riemann zeta function. But I am not getting the correct results when compared with the value of Wolfram alpha Hurwitz zeta ...
0
votes
0answers
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 $Y^{m+1}_n(\theta, \varphi)$...
1
vote
2answers
55 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?
2
votes
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{...
5
votes
2answers
200 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 $S_{n,p}(z)$, $$...
0
votes
1answer
871 views

Is $y(t) = x(2t)$ time invariant?

I'm confused on how to test this for time invariance. What is $x(t-t_0)$? Is it $x(2t-t_0)$ or $x(2(t-t_0))$? If it is the former, it would be considered time varying right? Because $x(2t-t_0)$ is ...
1
vote
1answer
94 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 ...
16
votes
1answer
373 views

Polylogarithm ladders for the tribonacci and n-nacci constants

While reading about polylogarithms, I came across the nice polylogarithm ladder, $$6\operatorname{Li}_2(x^{-1})-3\operatorname{Li}_2(x^{-2})-4\operatorname{Li}_2(x^{-3})+\operatorname{Li}_2(x^{-6}) = ...
0
votes
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 ...
1
vote
3answers
237 views

Integral involving Logarithm and Dilogarithm

I need your help in evaluating the following integral in closed form. $\displaystyle\int\limits_{0.5}^{1} \frac{\mathrm{Li}_{2}\left(x\right)\ln\left(2x - 1\right)}{x}\,\mathrm{d}x$ Since the ...
1
vote
1answer
28 views

Squared modulus of Dirichlet eta function.

I have a problem: Find formula for $|\eta(x+iy)|^{2}$, where $\eta(x+iy)=\sum_{n\ge1}\frac{(-1)^{n-1}}{n^{x+iy}}$ I calculated it in two ways and i got a contradiction. First method: $|\eta(x+iy)|...
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0answers
11 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.
0
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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.
11
votes
7answers
1k views

A gamma function inequality

I would like to prove $$\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)} \le \frac{1}{\sqrt{n}}$$ for all natural $n \ge 1$. The inequality does seem to be true numerically, but the proof eludes me.
1
vote
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 $...
2
votes
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}^...
0
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1answer
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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 $t$. $A_0$...
1
vote
1answer
711 views

Can anyone explain the inverse regularized beta function?

I'm working on a Wisdom of Crowds solution to be used in conjunction with a Genetic Algorithm for a class. The paper I am reading makes mention of an inverse regularized beta function, given by $I_{a_{...
1
vote
1answer
128 views

Evaluating $\int_0^1\frac{\ln(x)}{1-x/2}\int_0^{x/2}\frac{\ln(1-t)}{t}\,dt\,dx$

According to this post $$Q=\int_0^1\frac{\ln(x)}{1-x/2}\int_0^{x/2}\frac{\ln(1-t)}{t}\,dt\,dx=\frac{1}{8}\zeta \left( 4 \right) - \frac{1}{2}\zeta \left( 2 \right){\ln ^2}(2) + \frac{1}{{12}}{\ln ^4}(...
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votes
0answers
18 views

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
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 > ...
3
votes
2answers
162 views

Solution of $f(x)^2\frac{d^2}{dx^2}f(x)=x$

I am stuck in finding the solution of this apparently simple differential equation: $$f(x)^2\dfrac{d^2}{dx^2}f(x)=x$$ with$f(0)=a$ and $f(0)'=b$ Using Maple the solution seems to be a combination of ...
0
votes
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 ...
0
votes
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}...
2
votes
1answer
700 views

Approximating the log of the Modified Bessel Function of the Second Kind

I'm attempting to find a precise as possible approximation to the logarithm of the Modified Bessel Function of the Second Kind: $$\log K_{\alpha}(x) = \log\Big[\frac{1}{2}\int_0^{∞} t^{\alpha-1} \exp\...
2
votes
1answer
98 views

How to calculate singular moduli $\alpha_{3,n}$ of Ramanujan' s “$q_{3}$” theory?

Ramanujan's theory "$q_{3}$" How to calculate singular moduli $t=\alpha_{3,n}$ explicitly of this? $$\frac{\,_2F_1\big(\tfrac13,\tfrac23,1,1-t\big)}{\,_2F_1\big(\tfrac13,\tfrac23,1,t\big)} =\sqrt{n}...
1
vote
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 ) \...
0
votes
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 ...
0
votes
0answers
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,\...
1
vote
0answers
64 views

Embedding a Torus into Plane with Cuts

I have a complex plane with two horizontal cuts $[-\alpha \pm i/2, \alpha \pm i/2]$ for real $\alpha$. We can imagine gluing the two cuts to get a torus with complex parameter $\tau$. Thinking of ...
6
votes
1answer
563 views

Topology of Branch Cuts and Elliptic Integrals

In reading these notes (elliptic curves starting from elliptic integrals) I came across a couple claims about the topology of some complex surfaces. On page 4, they discuss the integral $$\phi(x) = \...
66
votes
3answers
4k views

Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \, \mathrm dx $ is in terms of the Barnes G-function. $$ \int_{0}^{z} \log \Gamma(x) \, \mathrm dx = \frac{z}{2} \log (2 \pi) + \...
4
votes
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 ...
0
votes
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 ...
0
votes
0answers
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,...
5
votes
0answers
159 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)...