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

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0
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0answers
43 views

Integration by parts with Bessel function $j_0$

I need to prove this: $$ \mathcal F{\frac{1}{r^2}}\frac{d}{dr}r^2 \frac{dC}{dr}$$ $$= (\frac{2}{\pi})^{1/2} \int_0^\infty\frac{1}{r^2}\frac{d}{dr}r^2\frac{dC}{dr}j_0(kr)r^2dr$$ $$ =-k^2 ...
6
votes
2answers
74 views

How to solve $\frac{\partial{\rm B}}{\partial b}\left(0^+,1\right)=-\frac{\pi^2}{6}$

Could you help me to prove $$\frac{\partial{\rm B}}{\partial b}\left(0^+,1\right)=-\frac{\pi^2}{6}$$ where ${\rm B}(a,b)$ is Beta function.
3
votes
0answers
36 views

Can derivative of Hurwitz Zeta be expressed in Hurwitz Zeta?

Can the derivative of Hurwitz Zeta function by the first argument be expressed in terms of Hurwitz Zeta and elementary fuctions? There is a formula which expresses Hurwitz Zeta through its ...
1
vote
1answer
36 views

Gamma of 3z using triplication formula:

I have to demostrate the gamma function for 3z as you see below: Using the multiplication formula demostrate gamma(3z) Gamma functions of argument $3z$ can be expressed using a triplication ...
8
votes
1answer
304 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: $$\int_0^1\frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx=\frac{\pi^2}{8}\zeta(3)-\frac{27}{16}\zeta(5) $$
0
votes
0answers
51 views

Show that this Hypergeometric Function equal to this gamma function

I have a question related to hypergeometric functions: Show that ...
5
votes
1answer
83 views

Pretty lower bound on the gamma function

According to http://functions.wolfram.com/06.05.29.0006.01, for every $x\geq 2$ it is $$ \left( \frac{x}{e}\right)^{x-1} \leq \Gamma(x) \leq \left( \frac{x}{2}\right)^{x-1}, $$ where $\Gamma$ is the ...
1
vote
0answers
31 views

Series in closed form

Let $\alpha \in (0,1)$ and $z\geq 0$. Define the function: $$ \theta_{\alpha}(z)=\sum_{j=0}^{\infty}\left(\frac{z}{\alpha j + 1}\right)^j. $$ Can $\theta_{\alpha}(z)$ be written in closed form? Are ...
4
votes
0answers
25 views

Function in Lipschitz space

I'm looking for a function that is in $W^{1,1}(0,1)$ but only in the Lipschitz space $\mathrm{Lip} (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $\mathrm{Lip}(\alpha, L_2(0,1))$ is defined as the set ...
2
votes
0answers
38 views

Simplified expression of $ _2F_1((K-1)a,K,Ka,x) $

Is there any simplified expression of this Hypergeometric function $ _2F_1((K-1)a,K,Ka,x) $ Thanks!
0
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0answers
17 views

How to determine singularities of a series?

Given a double Fourier series, how do we determine its singularities ? PS: I wonder how we find singularities(mathematically) if a function cannot be expressed in a closed form.
1
vote
1answer
52 views

Recognize as a special function?

Is the following function a special function of some kind $$ f(x) = \int_0^x (1+e^{-t})^{b}\,dt, $$ where $b>1$?
2
votes
1answer
41 views

Large-z limit of the *other* second derivative of the Laguerre polynomial

I'm trying to find the asymptotic behavior of the second derivative of the Laguerre polynomial (more precisely, the associated analytic function), $\frac{\partial}{\partial n^2}L_{n}(z)$, as $z\to ...
2
votes
0answers
45 views

A list of numbers and

I have a real life problem that math may be able to solve. I am no mathematician so if you have any insight please use the simplified version. This problem is way beyond me. My gut tells me there is ...
5
votes
1answer
90 views

Sum involving zeros of Bessel function

I came across the following sum in my work involving the infinite sum of function of zeros of Bessel functions. $$ \displaystyle ...
9
votes
0answers
225 views

Is this similarity just a coincidence?

Here is the function $-1/x$: If we add infinitely many similar functions with a shift of pi/2 each in both directions, we get $\tan x$. But if we do the same only in one direction, we get ...
2
votes
0answers
26 views

Identifying a function that involves combinations of terms

I need to know if a function exists that partitions terms in such a way as seen below $$ \frac{d^n}{dx^n}[\frac{(x)_c}{n!}] $$ Note that $(x)_c$ is the falling factorial of x and $c \geq n$, This in ...
0
votes
1answer
23 views

Monotonicity of Modified Bessel Functions of the Second type

Given $n\geq1$ an integer, Is it known that $$ x\to x^nK_n(x) $$ is a decreasing function on $(0,\infty)$? I am looking for a reference or a proof.
11
votes
1answer
172 views

An integral $\int^\infty_0\frac{\tanh{x}}{x(1-2\cosh{2x})^2}{\rm d}x$

I would like to enquire about the possible methods of computing the following integral $$ \color{blue}{% \int^{\infty}_{0}\frac{\tanh\left(\, x\,\right)} {x\left[\, 1 - 2\cosh\left(\, ...
1
vote
1answer
48 views

Solve ${y}' = \cosh^{-1}\left ( x \right ) + \mathrm {Si}(x)$

I am wondering how to find an explicit, closed-form solution for the following first-order differential equation: $${y}' = \cosh^{-1}\left ( x \right ) + \mathrm {Si}(x)$$ Where $\mathrm {Si}(x)$ ...
1
vote
2answers
107 views

Hypergeometric function integral representation

How to prove the following relation? $$ \, _2{F}_1(K,K;K+1;1-m) = \frac{\Gamma (K+1)}{\Gamma (K)} \int_0^{\infty } \frac{1}{(1+x) (m+x)^K} \, dx $$ where $_2{F}_1(.,.;.;.)$ is the hypergeometric ...
3
votes
1answer
43 views

Why does the whole integral converge but not part of it? (Dilogs)

$\newcommand{\Li}{\operatorname{Li}}$Consider the integral: $$\int_0^1 \frac{(-\Li_2(x) - \Li_3(x) - x^2/8 + 3x - x\log(1-x) + \log(1-x))}{x^2} \, dx$$ This integral converges to $\sim 0.01$ But ...
0
votes
0answers
78 views

Formula to divide (group) numbers into N proportionally groups

Lets take ideal theoretical case as example: we have 20 numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 we should split these numbers into 5 groups: lowest, low, ...
3
votes
0answers
73 views

Special functions, Fourier series

Well known are the Fourier expansions (presented, e.g., in Abramovitz and Stegun): $$ \cos ( A \sin x) = J_0(A) + 2 \sum_{k=1}^{\infty} J_{2k}(A)~\cos(2kx)~~, $$ $$ \sin ( A \sin x) = 2 ...
1
vote
0answers
26 views

Looking for a proof involving the Harmonic number [duplicate]

Prove that: $\displaystyle \sum_{k=1}^{\infty} \frac{H_k}{k^q} = (1 + \frac{q}{2})\zeta(q + 1) - \frac{1}{2}\cdot \sum_{n=1}^{q-2}\zeta(k+1)\zeta(q-k)$ It looks tough just to start off with. Any ...
6
votes
2answers
173 views

Closed form for a zeta series

It is not that diffcult to derive \begin{align} \sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{k2^k}=&-\frac{\gamma}{2}+\ln\left(\frac{2}{\sqrt{\pi}}\right)\tag{1}\\ ...
0
votes
0answers
52 views

Closed form of certain integral.

I am solving the following problem in heat transfer using the Laplace transform $$\rho\,c{\frac {\partial }{\partial t}}T \left( x,t \right) =k{\frac { \partial ^{2}}{\partial {x}^{2}}}T \left( x,t ...
1
vote
1answer
68 views

Where is the error in this proof :

Prove that: $$\frac {2\Gamma'(2z)}{\Gamma(2z)}-\frac {\Gamma'(z)}{\Gamma(z)}-\frac {\Gamma \prime(z+\frac{1}{2})}{\Gamma(z+\frac{1}{2})} =2 \log 2$$ But I obtain this equal zero. My proof: From ...
10
votes
3answers
525 views

How to integrate the dilogarithms?

$\def\Li{\operatorname{Li}}$ How can you integrate $\Li_2$? I tried from $0 \to 1$ $\displaystyle \int_{0}^{1} \Li_2(z) \,dz = \sum_{n=1}^{\infty} \frac{1}{n^2(n+1)}$ $$\frac{An + B}{n^2} + ...
13
votes
1answer
249 views

Integral that arises from the derivation of Kummer's Fourier expansion of $\ln{\Gamma(x)}$

I am trying to prove that for $0<x<1$, $$\color{blue}{\ln{\Gamma(x)}=\frac{1}{2}\ln(2\pi)+\sum^\infty_{n=1}\left\{\frac{1}{2n}\cos(2\pi nx)+\frac{\gamma+\ln(2\pi n)}{n\pi}\sin(2\pi ...
3
votes
0answers
53 views

Closed form of a “harmonic” alternating dilogarithm sum

Does the following sum $$ S = \sum_{n\geq 2}(-1)^n \mathrm{Li}_2(2/n) = 1.14434\ 42096\ 91982\ 23727\ 39852\ 45805\ldots $$ have a closed form in terms of known constants? Neither the inverse ...
2
votes
1answer
54 views

Question about $\frac {\Gamma'(z+1)}{\Gamma(z+1)}$

If $\psi (z)= \log\Gamma(z+1)$ Prove that : $$\psi(n)+\gamma=1+\frac{1}{2}+\cdots+\frac{1}{n}$$ My Proof : $$\psi (z)= \frac {\Gamma'(z+1)}{\Gamma(z+1)}=-\frac{1}{z+1}-\gamma + \sum_{n=1}^\infty ...
1
vote
1answer
76 views

Special Integral Proof

How to prove $$\int_0^\infty x^{2n-1} \exp(-a^{x^3})\, dx = \frac{\Gamma(n)}{2a^n} ,\quad n> 0 ,\quad a>0. $$
1
vote
1answer
45 views

Question on series $\frac {\Gamma'(z)}{\Gamma(z)}$

Prove that: $$\frac {2\Gamma'(2z)}{\Gamma(2z)}-\frac {\Gamma'(z)}{\Gamma(z)}-\frac {\Gamma \prime(z+\frac{1}{2})}{\Gamma(z+\frac{1}{2})} =2 \log 2$$ But I obtain this equal zero: $$\frac ...
1
vote
1answer
47 views

How can we solve the “transcendent” equation relating to Stoner criterion

I met a algebraic equation(not a transcendent equation) during my study of Stoner criterion in Quantum Statistical Physics. In this occasion, one need to solve the equation $$ ...
2
votes
1answer
32 views

Sum of complex digamma functions

It seems that the sum of the digamma function of $z$ and the digamma function of its conjugate $z^*$ is always real-valued. ...
1
vote
0answers
31 views

Ellptic\Jacobi theta function and its residue integral

The Ellptic\Jacobi theta function is given by \begin{align} \theta_1(\tau|z)&=\theta_1(q,y)=-iq^{\frac{1}{8}}y^{\frac{1}{2}}\prod_{k=1}^{\infty}(1-q^k)(1-yq^k)(1-y^{-1}q^{k-1}) \\ &= ...
0
votes
0answers
29 views

q-theta function and their properties

I want to compute the residue integral for q-theta function, and derive its properties. First i'll briefly explain the definition \begin{align} & ...
0
votes
0answers
30 views

Identities related to hypergeometric functions

It is known that hypergeometric functions are closely related to the formula of $\pi$ given by Ramanujan. Trying to master the proof given by the Borwein brothers, I got two identities: ...
0
votes
1answer
46 views

Differential equation with zero solution of indicial equation?

I want to solve this equation $$ y'' + (\frac{1}{x} + 4x)y' + (5+4x^2)y = 0 $$ Where $y''$ is second derivative and so on. This equation has singuar point at $x=0$. And this is regular singular ...
5
votes
0answers
126 views

Integral of a product of five Bessel functions of order $0$

Does the following integral have a closed form? $$ \mathcal{J}(2,3,5,7,11) = \int_0^\infty x J_0(x\sqrt{2})J_0(x\sqrt{3})J_0(x\sqrt{5})J_0(x\sqrt{7})J_0(x\sqrt{11})\,dx. $$ I know that some similar ...
0
votes
0answers
20 views

How to find this limit of Bessel function?

I have a question about limit of Bessel function. $$ \lim_{x \to \infty} x \bigg [ (J_p (x) )^2 + (Y_p (x) )^2 \bigg ] $$ Where, $ J_p (x)$ is Bessel function of first kind $ Y_p (x)$ is general ...
2
votes
1answer
93 views

how to integrate $\mathrm{arcsin}\left(x^{15}\right)$?

Integral by parts: $$ I = x\sin^{-1}\left(x^{15}\right) - \int\frac{15x^{15}}{\sqrt{1-x^{30}}}dx $$ then what? The answer by wolfram gives an answer contains hypergeometric ${}_2F_1$ function,because ...
2
votes
0answers
47 views

Asymptotics of inverse Laplace transform of a function with an essential singularity?

Let $h$ be the function $$ h(x) = \sum_{k\geq0} \frac{(ix)^k}{k!}\zeta(2k), $$ with the Laplace transform $$ \tilde h(s) = -\frac{\pi}{2s}\sqrt{i/s}\cot\left(\pi\sqrt{i/s}\right), $$ which has an ...
0
votes
0answers
14 views

Is there such a thing as a “continuum singular value decomposition”?

I have a question about expressing 2D functions as sums of separable functions. As a concrete example, consider the Gaussian circle function, ...
1
vote
0answers
32 views

Does a specific function exist with these properties?

lets say i have a function where: $[p,c,n,k] \in \mathbb{Z}$ defined some way in this function $ f(x) = \sum_{q=0}^{n} \sum_{v=1}^{k-q+1}\frac{c(k+1-v)!(p-k)!}{(k+1-v-q)!q!(p-k-q)!}x^{k-v+1-q} = ...
1
vote
1answer
41 views

Showing that the linear twist map is sensitive dependent

Choose $\Delta=\frac{1}{2}$ (I believe this value should work). let $\delta > 0$ and let $\textbf{x}_1=(x_1,y_1) \in X$. I assuming that $d$ is the Euclidean distance. Somehow I think we ...
7
votes
1answer
193 views

Is this integral reducible to an elliptic integral?

I believe if $k=0$ the following integral is reducible to an elliptic integral. If $k > 0$ is it possible to reduce it to an elliptic integral or some other special function? $$\int_\rho^x \sqrt{1 ...
6
votes
1answer
155 views

Prove this closed-form of sum of ${_4F_3}$ hypergeometric functions

I think the following identity is true. How could we prove it? $${_4F_3}\left(\begin{array}c 1,1,1,1 \\\tfrac54,2,2\end{array}\middle|\,1\right) + ...
1
vote
1answer
51 views

Is this a special function?

Suppose $$ f(z;a) = \int_0^z t^{-a-1}\,(1+t)^{a}\,dt, $$ where $a>1$. Is this function known as a special function? It appears to be close to the following representation of the beta function: $$ ...