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

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2 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!
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0answers
12 views

For an open local-injective map f(∂U) ⊆ ∂f(U)

let U be a subset of R^n. f is a continues open, locally-injective function f:U --> R^k. I know that for an injective map, f(∂U) ⊆ ∂f(U). I think it should also be the case with a locally-injective, ...
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0answers
8 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.
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1answer
36 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 $b>1$?
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0answers
15 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), $\frac{\partial}{\partial \nu^2}L_{\nu}(z)$, as $z\to ...
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0answers
39 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 ...
3
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0answers
39 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. Does anyone have any idea of how to express this in a simpler form ? or at least an ...
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0answers
130 views
+50

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 ...
-3
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38 views

[Find the root of the equation, please help] [closed]

I have a problem need your help, I have the following equation$$x\ln(x)+ax^2 + bx + c = 0.$$ It look like Lambert function, but I cannot find the root of this equation. Everybody can help me to find ...
2
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0answers
24 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 ...
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1answer
18 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.
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1answer
110 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 $$\color{blue}{\int^\infty_0\frac{\tanh{x}}{x(1-2\cosh{2x})^2}{\rm d}x=\ ?}$$ A possible way I see of doing this ...
1
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1answer
41 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
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2answers
88 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
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1answer
38 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 ...
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0answers
33 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, ...
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0answers
56 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 ...
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0answers
23 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 ...
5
votes
2answers
126 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}\\ ...
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0answers
44 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
60 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 ...
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3answers
498 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
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1answer
187 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 ...
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0answers
36 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
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1answer
49 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
70 views

Special Integral Proof

How to prove $$\int_0^\infty x^{2n-1} \exp(-a^{x^3})\, dx = \frac{\Gamma(n)}{2a^n} ,\quad n> 0 ,\quad a>0. $$
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1answer
41 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 ...
0
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1answer
23 views

finding minima of three varible function

I need to find minima of this function. $f(a,b,c)=2^a-5^b\cdot7^c$ where $a,b,c$ are positive integer I need to prove that for any value of a,b,c the value of function can never be 1. Tried ...
1
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1answer
34 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 $$ ...
1
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1answer
18 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. ...
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0answers
12 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}) \\ &= ...
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0answers
7 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} & ...
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0answers
25 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: ...
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1answer
29 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
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0answers
111 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 ...
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0answers
15 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
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1answer
80 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
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0answers
28 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 ...
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8 views

Hilbert transform of the product of functions

Let $H[g]$ denotes a Hilbert transform of function $g$. What would be the constant $C$ in the following inequality: $$ \|H[(\cos n)(\cos{1/(2n))}f](x)\|_{L_2}\leq C\|f\|_2? $$
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10 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, ...
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0answers
31 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} = ...
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1answer
28 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
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1answer
162 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
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1answer
136 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) + ...
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1answer
47 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: $$ ...
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0answers
21 views

What is a “hypergeometric series” with differences, not just sums, of indices?

"Hypergeometric series" often have forms like (in two variables) $$\sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{(a_1)_n (a_2)_n}{(b_1)_k (c_1)_{n+k}} \frac{x^n}{n!} \frac{y^k}{k!}$$ And there are ...
5
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0answers
80 views

Closed-forms of real parts of special value dilogarithm identities from inverse tangent integral function

The inverse tangent integral is defined as $$\operatorname{Ti}_2(x)=\Im\operatorname{Li}_2\left(ix\right)$$ Because this we have some special value identitiy. Let $c_1 = \operatorname{Li}_2(i)$, ...
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0answers
13 views

Incomplete Beta function for negative parameters

I implemented the incomplete Beta function $B_x(a,b)$ for negative $a,b$ using the relations to the Hypergeometric function from http://functions.wolfram.com/GammaBetaErf/Beta3/26/01/02/, especially ...
3
votes
2answers
55 views

Decomposition of $_1F_2(1+n;1,2+n;x)$

I am looking for a way to decompose $_1F_2(1+n;1,2+n;z)$ for $n\in\mathbb{N}$ into either Bessel J functions or regularized confluent hypergeometric functions $_0\tilde F_1(b(n),z)$. Mathematica seems ...
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0answers
21 views

Name for hypergeometric-like sum.

Consider the sum $F(a_1,\cdots,a_r; z)\sum_{m=0}^{\infty}\frac{(a_1+m)(a_2+m)\cdots(a_r+m)}{(a_1)(a_2)\cdots (a_r)}\frac{z^m}{m!},$ where $a_i$ for $i=1,2,\cdots,r$ is an increasing sequence of ...