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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52 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
0
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0answers
33 views

Evaluation of $\int_{1}^{\infty} x^{-\frac{5}{3}} \cos \left( \left( x-1 \right) h \right)dx$ with Maple

I have calculated the Integral with the aid of some professors here and I get a problem: $$\int_{1}^{\infty} x^{-\frac{5}{3}} \cos \left( \left( x-1 \right) h \right)dx$$ I have done the Integral ...
0
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0answers
18 views

Inequality involving special functions

Denote by ${}_2F_1$ the hypergeometric function (see https://en.wikipedia.org/wiki/Hypergeometric_function). Let $n,m\in\mathbb{Z}$ s.t. $2\leq m\leq n$, $$f_k\left(x\right) \equiv ...
2
votes
2answers
51 views

Skewed Trigonometric Function

What would be an expression for a periodic function (period $2\pi$) that essentially behaves just like a negative sine function, but it has the following quirk: It's $0$s lie on the usual places ...
2
votes
1answer
44 views

Application of Bessel Function

I have read number of books and online literature on Bessel function. Theoretically, I have known about Bessel function. What is practical significance of Bessel function? How can Bessel function ...
4
votes
1answer
55 views

Spherical integral

Let $y \in \mathbb{R}^n$ be fixed. Is there a nice expression for the following integral taken over the unit sphere in $\mathbb{R}^n$? $$ \int_{\|x\|=1} e^{2\pi i (x \cdot y)}~dx $$
0
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0answers
20 views

What is the asymptotic behaviour of $n^3 \log(\Gamma(1 + 1/n))$ as $n\to\infty$?

What is the asymptotic behaviour of $n^3\log(\Gamma(1 + 1/n))$ as $n\to\infty$ ? I have deduced that $\log\Gamma(1+1/n)\sim\log(1-\gamma/n)$ but multiplying by $n^3$ then gives an error.
0
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0answers
26 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
-1
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0answers
18 views

The spectral density of this signal

I want to calculate the spectral density of $m(t) = (a(k)-a(k-1))\times j(k) \times p(t-kT)$, where, $T$ is the bit period, $k$ is one point in time, $a(k) \in {\pm 1}$ and are equiprobable and $j$ is ...
13
votes
1answer
162 views

A Gamma limit $\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$

Show that $$\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$$ where $\gamma$ is the Euler-Mascheroni Constant. ...
0
votes
1answer
21 views

multiderivative of incomplete gamma function

I spend 4 hours trying to solve it. I believe that I am either stuck or I am approaching the problem at the wrong angle. Here is my challenge: $$\frac{\partial^m }{\partial x^m}\left [ x^{-a}\gamma ...
3
votes
1answer
76 views

Compute the following series $\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}$

Does the following series have a 'closed' form : $$\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}.$$ Where $n\in \Bbb{N}$ and $a,b \in (0,+\infty)$ For $a,b$ integer we can use Partial fraction ...
8
votes
2answers
83 views

Trigamma identity $4\,\psi_1\!\left(\frac15\right)+\psi_1\!\left(\frac25\right)-\psi_1\!\left(\frac1{10}\right)=\frac{4\pi^2}{\phi\,\sqrt5}.$

I heuristically discovered the following identity for the trigamma function, that I could not find in any tables or papers or infer from existing formulae (e.g. [1], [2], [3], [4], [5], [6]): ...
3
votes
1answer
41 views

Find zero of sum of 4 modified Bessel functions

I am trying to find the (positive) root of the function $f(x) = I_{-3/4}(x) + I_{3/4}(x) - I_{-1/4}(x) - I_{1/4}(x)$ where $I_\alpha(x)$ denotes the modified Bessel function of the first kind. ...
0
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0answers
92 views

Why is it hold for types of operators?

We ‎let ‎the ‎state ‎space ‎be‎ ‏‎‎‎$ \mathcal{H} =‎ ‎‎H_{E}^{2}(0 , 1) \times L^2(0 , 1) $‎ equipped with the norm ‎ \begin{align} \| (f , g) \| = \int_{0}^{1} [ |f''(x)|^2 + |g(x)|^2] ‎\mathrm{d}x‎ ...
8
votes
1answer
73 views

Closed form for integral of integer powers of Sinc function

(Edit: Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.) For all $n\in\mathbb{N}^+$, ...
2
votes
1answer
30 views

Uniqueness proof for a certain functional equation

Consider the following functional equation: \begin{equation}f(x)=kf(mx)\end{equation} where $x \in [0,1]$; $k>0$; $0<m<1$; and $f(1)=1$. It has at least one solution: $f(x)=x^l$, where ...
3
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0answers
18 views

integral over product of two bessel functions at discontinuity

The Weber-Schafheitlin integral $$ \int_{0}^{\infty}\frac{J_{\mu}(a t)J_{\nu}(bt)}{ t^{\lambda}} $$ where $J_{\mu}(x)$'s are Bessel functions of the first kind, can have delta function singularities ...
1
vote
3answers
46 views

Bounding a modified Bessel function of the first kind.

Let $I_0$ be the zeroth-order modified Bessel function of the first kind. We know that, asymptotically as $x\to \infty$, $I_0(x) \sim e^x/\sqrt{2\pi x}$. Does anybody have a reference for the maximum ...
2
votes
1answer
74 views

Looking for an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$

I have a recollection of seeing an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$. But I don't remember details, and searching my books and the Internet ...
2
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1answer
31 views

Computing of the Gamma Function

I have stumbled upon Gamma functions when dealing with Gamma distributions on my studies with basic statistics. However, I have not understood how its computation expands factorials to real and ...
3
votes
2answers
141 views

Closed form for $\int_0^x \{1/t \}\,\mathrm{d}t$, $x \in \mathbb{R}_+$ and related.

After some tests I think that Conjecture 1 Let $x \in \mathbb{R}_+$ then $$ \int_0^x \left\{ \frac{1}{t} \right\}\,\mathrm{d}t = 1 - \gamma + H_{\{1/x\}} - x\lfloor1/x\rfloor + \log x$$ ...
0
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0answers
35 views

Hyperbolic sinc function

Cardinal sine function or sinc function is defined by: \begin{equation} \mathrm{sinc}x=\begin{cases}\frac{\sin x}{x}, & x \neq 0,\\ 1, & x = 0,\end{cases} \end{equation} Is there any ...
1
vote
0answers
18 views

Uniform perturbative solutions to the Mathieu equation

The Mathieu equation is a second-order linear differential equation given by $$y''(t) + [a - 2q\cos(2t)]y(t) = 0$$ There are two special functions defined as linearly independent solutions to ...
1
vote
1answer
46 views

Differentiation of the Beta function

I suppose that \begin{align*} \frac{\partial}{\partial x}\left[B\left(x,y\right)\right]=&\frac{\partial}{\partial x}\left[\int_0^1t^{x-1}(1-t)^{y-1}dt\right]\\ ...
0
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0answers
15 views

What is the equation for an anisotropic Hanning window (cosine wave) in two or three dimensions?

I do not exactly know how to ask this question, so I will explain myself thoroughly. I am really stuck on this one, and it is crucial for my research, so if anyone has any ideas on where I may find ...
3
votes
1answer
91 views

what does the ${_2F_1}\left[\cdot, \cdot, \cdot, \cdot\right]$ function mean?

I was reading the integral tables, where it says: \begin{equation} \int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times {_2F_1}\left[ \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax ...
1
vote
0answers
67 views

Verification of Fourier transformation of Io-sinh function

I try to match, but it could not match $I_o-\sinh$ Practical Fourier Transform pair developed by Ben Logan, transform pair also published in The Practical Application of the Fourier Integral ...
0
votes
0answers
30 views

Simplify $L_{-1}(x) + I_1(x) $

Is there a simple solution for x << 0 of the following equation: $$Y(x) = L_{-1}(x) + I_1(x) $$ Where $L_{-1}(x)$ is modified Struve function and $I_1(x)$ is modified bessel function. For ...
6
votes
1answer
63 views

A generalization of Bell numbers to arbitrary complex arguments

For $n\in\mathbb N$, the Bell number $B_n$ is a number of ways to partition the integer range $[1,\,n]$ into pairwise disjoint non-empty subsets. E.g. $B_3=5$ because ...
9
votes
2answers
278 views
6
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3answers
177 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
2
votes
1answer
62 views

Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $$I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta,$$ where $a$ and $b$ are real numbers. If we ...
8
votes
1answer
90 views

Prove $_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0$

Please help me to prove the identity $$_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0,$$ where $\phi=\frac{1+\sqrt5}2$ is the golden ratio.
0
votes
1answer
22 views

Even function about a point over a restricted range

Why is $f(x)=(x-1)^2$sin$(n\pi x)$ even about $x=1$ for $0\leq x \leq2$? I understand that $(x-1)^2$ is even about $x=1$ and I can plot the graph for various values of $n$ on wolfram alpha, but how ...
10
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0answers
165 views
1
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2answers
45 views

Help with operator $f(x^q)=\frac{1}{q+1}x^q$.

This question is somewhat related to this. I am looking for an operator $f:\mathbb{R}[x]\to\mathbb{R}[x]$, that is, $f$ is an operator that maps polynomials in one variable to polynomials in one ...
16
votes
2answers
328 views

Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

I discovered the following conjecture numerically, but have not been able to prove it yet: $$_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47.\tag1$$ The equality holds ...
0
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0answers
47 views

Integral of Hypergeometric Function with polynomial and exponential

I was working on some mathematical derivations and faced this integral: $$I=\int_{0}^{\infty}x^{\alpha-1}e^{-\beta x}{_2F_1}{(a,b;c;1-hx)}\,\mathrm{d}x$$ how can I integrate it?
14
votes
1answer
261 views

Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

I discovered the following conjectured identity numerically (it holds with at least $1000$ digits of precision). How can I prove it? ...
1
vote
0answers
47 views

Integrating $\int_0^1 dx\,\ln(x-a)/(x-b)$ paying attention to cuts.

I am trying to compute the following integral, for complex $a$ and $b$ $$\int ^1 _0 dx \frac{\ln(x-a)}{x-b}$$ by turning it into something in terms of dilogarithms. But for certain values of $a$ ...
2
votes
2answers
43 views

Sum of odd Bessel Functions

In an answer, I showed that: $$\sin(1)=2\sum_{k=0}^\infty(-1)^k J_{2k+1}(1)$$ Where $J_n(x)$ is the Bessel function of the first kind. Is there a more general result for the infinite sum of odd Bessel ...
1
vote
0answers
16 views

Basis representation for non-negative, compact support, reasonably smooth spectral function

I was wondering if anyone has ideas on representing a non-negative, compact support (from x=-1 to 1 on the real axis) spectral function as a superposition of basis elements. Ideally, the basis ...
3
votes
1answer
60 views

What does this paper mean by “$f(x)$ is practically a rational function”?

The paper "Infinite integrals involving Bessel functions by contour integration" by Qiong-Gui Lin gives a method to solve integrals of the form $\intop_{0}^{\infty}x^{v}f(x)J_{v}(qx)\, dx$. One of the ...
1
vote
2answers
41 views

Help with function $f_r(x^q)=q^rx^{q-1}$

Let $r,q$ be a positive integers. I am looking for a function $f_r(x^q)$ such that it is satisfied $$ f_r (x^q)=q^r x^{q-1}$$ (without explicit dependence on $q$ of course, and for $r>1$). I ...
0
votes
1answer
47 views

Pair of functions $F(x)$ (transcendental),$A(x)$ (algebraic) with expanded series of positive integer coefficient linked by derivative

$$F(x)=\sum_0^{\infty}b_k x^k,b_k\in \mathcal{N} \bigcup 0,\exists M \space b_k \leq M^k$$. $$A(x)=\sum_0^{\infty}a_k x^k,a_k\in \mathcal{N} \bigcup 0,\exists M \space a_k \leq L^k$$ where $F(x)$ is ...
6
votes
1answer
120 views

How do people on MSE find closed-form expressions for integrals, infinite products, etc?

I always wanted to ask this question since when I joined MSE, but because I was afraid of asking too many soft questions I never asked it. I've seen some pretty complicated integrals and infinite ...
4
votes
2answers
155 views

Convergence of ${\large\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx$

Consider $$I(a)={\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx,$$ where $J_0(z)$ is the Bessel Function of the $1^{st}$ kind and $a>0$. Does this integral converge for any values of $a$? If so, is ...
7
votes
1answer
99 views

Derivative of a generalized hypergeometric function

Let $$f(a)={_2F_3}\left(\begin{array}c1,\ 1\\\tfrac32,\ 1-a,\ 2+a\end{array}\middle|-\pi^2\right).$$ How to find $f'(0)$ in a closed form?
2
votes
2answers
38 views

Evaluation of Indefinite Integral resulting in Hypergeometric Function

I am attempting to derive the result: $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$ First, I start off with the binomial expansion of the integrand to ...