Questions tagged [closed-form-expressions]

For questions that specifically ask for determining a closed form of equations, integrals etc.

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0answers
47 views

Extension of closed-form solvability of polynomial equations of x and exp to rational expressions?

Is my conjecture below true? It's a new conjecture. It's an extension of Lin's theorem in [Lin 1983] which is cited in [Chow 1999] - see both references at the bottom of this page. It seems that Ferng-...
2
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1answer
152 views

From Zurab's integral representation for the Apéry's constant to almost impossible integrals

I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...
3
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1answer
228 views

A second order nonlinear ODE

In my research (in differential geometry) I recently came across the following nonlinear second order ode: $$\frac{f''(x)}{f'(x)}-\frac{2}{x}+\frac{f'(x)+1}{2f(x)-x-1}+\frac{f'(x)-1}{2f(x)+x}=0$$ It ...
2
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0answers
110 views

How do I evaluate the following double integral?

I would like to evaluate the following double integral: $$ \int_{-1}^1d\zeta\int_{-1}^1 d\bar{\zeta} (\zeta+\bar{\zeta})^{d-2}[(1+\zeta\bar{\zeta})(\zeta-\bar{\zeta})]^J \,\times [(1-\zeta)(1+\bar{\...
1
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0answers
63 views

Why do we not have a closed form expression for counting transitivity?

https://en.wikipedia.org/wiki/Transitive_relation. Are there any theoretical reasons out there which show us that why do we still not have a closed-form expression for transitivity counting. If you ...
1
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1answer
290 views

Find closed-form expression to $f(n)$

Let $ \forall n\in\mathbb N.\quad f(n)= \begin{cases} \min_{a\in\{\lceil n/2\rceil, \lceil n/2\rceil+1,\dots, n-1\}} \frac 1 4 \binom n a f(a) & \text{if $n\geq 4$}\\ 1 & \text{else} \end{...
8
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2answers
373 views

Supremum of $ a_n = a_{n-1}^3 - a_{n-2} $ [closed]

Let $a_1=0$ and let $ - \ln(2) < a_2 < \ln(2) $ Define $$ a_n = a_{n-1}^3 - a_{n-2} $$ Then $$ \sup_{n>2} a_n = a_2 $$ And $$ \inf_{n>2} a_n = - a_2 $$ How to prove that ?
0
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1answer
105 views

PDF of $z = \exp(j\varphi)$, where $\varphi \sim \mathcal{U}[-a, +a]$ [closed]

How can I find the PDF of $z = \exp(j\varphi)$, where $\varphi \sim \mathcal{U}[-a, +a]$, $i.e.$, a uniformly distributed r.v.? My difficulty here is that it involves complex numbers and I don't know ...
2
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0answers
74 views

Closed form expression for this Fourier summation?

Consider the function $f:\mathbb{T}^m\to\mathbb{R}$ $$f(\boldsymbol{x}) = \sum\limits_{{\pmb{\eta}\in\mathbb{Z}^m}}\frac{1}{1+\lambda\|\pmb{\eta}\|_{2k}^{2k} } \cos({2\pi \pmb{\eta}\cdot\pmb{x}})$$ ...
1
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2answers
284 views

Closed form of $\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$

I have made a question here about closed form of the following: $$\prod_{k=0}^{N}\big(k!\big)^{{N}\choose{k}}$$ I know that there is a known closed form for, $$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$ ...
4
votes
1answer
291 views

About $a_n = \frac{a_{n-1}(a_{n-1} + C)}{a_{n-2}}.$

Let $A,B,C > 0$. Put $a_1 = A$ and $a_2 = B$ and, for integer $n > 2$, $$a_n = \frac{a_{n-1}(a_{n-1} + C)}{a_{n-2}}$$ and $$ T = \lim_{k \to \infty} \frac{a_k}{ a_{k - 1}}.$$ Notice the limit ...
10
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1answer
343 views

Closed Form for $\sum\limits_{n=-2a}^\infty(n+a){2a\choose-n}^4,~a\not\in\mathbb Z$

Do either $~S_4^+(a)~=~\displaystyle\sum_{n=0}^\infty(n+a){2a\choose n}^4~$ or $~S_4^-(a)~=~\displaystyle\sum_{n=-2a}^\infty(n+a){2a\choose-n}^4~$ possess a meaningful closed form expression1 in terms ...
2
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2answers
343 views

PDF of $ | \sum_{k=1}^{n}{|h_k||g_k|\exp\left( j \theta_k \right)} |^2$ for small values of $n$ and $Q$?

Given the following function of random variables $$f = \left|\sum_{k=1}^{n}{|h_k||g_k|\exp\left( j \theta_k \right)} \right|^2,$$ where $h_1, \cdots, h_n$ and $g_1, \cdots, h_n$ are i.i.d. random ...
0
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2answers
136 views

PDF of $g = \frac{1}{n} \sum_{k=1}^{n}{|h_k|\exp\left( j \theta_k \right)}$?

Given the following function of random variables $$g = \frac{1}{n} \sum_{k=1}^{n}{|h_k|\exp\left( j \theta_k \right)},$$ where $h_1, \cdots, h_n$ are i.i.d. random variables following the complex ...
0
votes
2answers
212 views

PDF of $R$ given that $R^2 = C^2 + S^2$, with $C = \sum_{j=1}^{n}{\cos \theta_j}$ and $S = \sum_{j=1}^{n}{\sin \theta_j}$ for a small $n$

Suppose that $\theta_1, \cdots, \theta_n$ are distributed independently and that $\theta_j$ has probability density function (PDF) $f_j = \frac{1}{2\pi}$ ($i.e.$, the uniform distribution) for $j = 1, ...
6
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0answers
174 views

A class of symmetric functions

When attacking a symmetric problem via an asymmetric method, I encountered the following function: $$U_2(n, m) = \sum_{a = 0}^n\binom na (2^a + 2^{n - a})^m.$$ It is easy to see that this function is ...
0
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0answers
62 views

What is the closed-form solution to this double-sum norm function?

Given two points $A,B \in \mathbb{R}^2$, one defines the Euclidean distance $f: \mathbb{R}^2\times\mathbb{R}^2 \rightarrow \mathbb{R}^{\ge 0}$ as follows. $$f(A,B) := \Vert A-B \Vert : = \sqrt{(A_{x}...
2
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0answers
161 views

Proof of non-solvability of general equations of one unknown in elementary terms (finite terms)?

This question relates to the solvability of equations of one unknown in elementary terms (finite terms) according to Liouville and Ritt. Elementary equations and closed-form solutions can be ...
2
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0answers
73 views

Closed form for unusual recurrence

We have for $k>0$, $n>0$, $m\geqslant0$ $$p_k(n,m)=k(n-1)!\sum\limits_{s=0}^{n-1}\frac{p_{k-1}(s+1,m+1)+p_{k-1}(m+1,s)}{s!}$$ also $$p_0(n,m)=\begin{cases} (n-1)!,&\text{$n>0, m=0$}\\ 0,&...
1
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0answers
417 views

The mysterious numbers $ \frac{13}{20} $ and $20$?

Let $g(x) = x^6 - 30 x $ Let $h(x) = x^6 $ Let $f(x) = x^2 - 2 $ Let $r$ be a reduced fraction $0 < \frac{p}{q} < 2 $ with integers $p,q > 1$ Let $f_{n+1}(x) = f(f_n(x)) = f_n(f(x)) , ...
4
votes
1answer
321 views

On the integral $\int_0^1\log(x!)dx$ revisited

I was interested in an integral that I known from [1], it is $$\int_0^1 \log(x!)dx.$$ I tried to get such closed-form using myself ideas and symbolic calculations, also with the help of Wolfram ...
1
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2answers
64 views

Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians?

Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
1
vote
1answer
305 views

On $\zeta(7)$ as the integration of the product of an indefinite integral due to Lobachevskii by a power of the inverse Gudermannian function

In this post I invoke certain function from a post of this site MathOverflow it is [1] (please see further references from the post, authors from the Springer link of the cited literature and answers ...
2
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0answers
203 views

Closed form expression for $Tr\left[ (\mathbf{DW})^k \right]$

Given the $N \times N$ diagonal matrices $\mathbf{D}$ and $\mathbf{W}$ as defined below $ \begin{split} \mathbf{DW} &= \left[ \begin{array}{cccc} \beta_{1} & 0 & \cdots & 0 \\ ...
4
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0answers
601 views

Why are there elementary equations that are not solvable in closed form?

Elementary equations and closed-form solutions can be important in mathematics and in the natural sciences, e.g. for discovering and describing certain relationships. $\log\colon x\mapsto\log(x)$; $x\...
22
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1answer
1k views

Why these surprising proportionalities of integrals involving odd zeta values?

Inspired by the well known $$\int_0^1\frac{\ln(1-x)\ln x}x\mathrm dx=\zeta(3)$$ and the integral given here (writing $\zeta_r:=\zeta(r)$ for easier reading)$$\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm dx=...
0
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0answers
49 views

Is the exact solution of the wave equation for the scattering of waves by a disk/cylinder an open problem?

The solution exact solution of the Helmholtz equation for the scattering of waves by a sphere is relatively straightforward and has been known since the time of Lord Rayleigh. The exact solution of ...
5
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0answers
121 views

Conjecture for a certain Cauchy-type determinant

Given the Cauchy-like matrix $$ \mathbf X_M(q) = \left[ \frac{2}{\pi} \frac{ \Gamma\!\left(m - \frac{1}{2} \right)\Gamma\!\left(n + \frac{1}{2} \right) }{ \Gamma(m)\,\Gamma(n) } \frac{m-\frac{3}{4}} {\...
-2
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1answer
170 views

About infinite products and Euler Gamma functions [closed]

I am interested in knowing how to calculate infinite products like (or reading any reference about it): $$\prod_{j=1}^{\infty}\left( 1-\left( \frac{x}{a+j\pi} \right) ^2 \right)$$ Inserting it into ...
0
votes
1answer
63 views

CDF of a RV that is the ratio between a complex Gaussian and a Chi-squared RVs

Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs: \begin{equation*} f_U(u)=\exp\...
2
votes
2answers
130 views

Expectation of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\textbf{x}$ and $\textbf{y}$ are complex Gaussians?

Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
2
votes
1answer
83 views

p.d.f. of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\textbf{x}$ and $\textbf{y}$ are complex Gaussians?

Given that the random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ are ...
2
votes
2answers
223 views

Closed expression for $\mathbb{E} \left\lbrace \Re \frac{(\textbf{x} + \textbf{y})^{H}\textbf{x}}{\| \textbf{x} + \textbf{y}\|^{2}} \right\rbrace$?

Given the random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ are independent, ...
3
votes
0answers
231 views

How to extend Ritt's theorem on elementary invertible bijective elementary functions?

The elementary functions of a complex variable $z$ according to Liouville and Ritt are those functions built up from the rational functions of $z$ by exponentiation, taking logarithms, and algebraic ...
1
vote
4answers
465 views

how to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$ [closed]

I’d like to solve following equation for $x.$ If it is not possible, why I can’t? $$\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$$
4
votes
1answer
143 views

Asymptotics for sum involving Euler numbers

This first request may be easy, but the asymptotics for the next step has me scratching my head. Through an informal inductive argument I have been able to show $$ (1) \quad \sum_{j=0}^{n-1}2^{2m(n-j)...
3
votes
2answers
257 views

Infinite sum of reciprocals of squares of lengths of tangents from origin to the curve $y=\sin x$

This question is actually from MSE. I had to post it here due to the lack of response there even after placing a bounty. Here goes the question Let tangents be drawn to the curve $y=\sin x$ from ...
1
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0answers
385 views

Closed-form formula for Wasserstein distance between uniform discrete distribution and discrete distribution with same support

Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{...
4
votes
1answer
387 views

Approximation a sum involving log and binomial coefficient

I am wondering about the asymptotic approximation of the following expression: $$S=\sum^{N}_{i=0}\log\Bigg[\binom{\binom{N+1}{i}}{t_i}\Bigg]$$ where $$t_i=\binom{N}{i}-\binom{N-k}{i-k}+\binom{N-k}{i-...
10
votes
2answers
1k views

Difficult trigonometric integral

This question was also asked here and here. I have faced some difficulties to do the following integral: $$ I=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta~\sin\theta\int_{0}^{\infty}dr~r^2\frac{3x^2y^...
7
votes
4answers
912 views

Find a formula for the recurrent sequence $q_{n+1}=q_n(q_n+1)+1$

Find an analytic formula for the recurrent sequence $$q_{n+1}=q_n(q_n+1)+1,\;\;q_0\in\mathbb N.$$ (The question was asked on 03.05.2018 by M. Pratsovytyi, see page 109 of Volume 1 of the Lviv ...
1
vote
1answer
235 views

On finding the critical points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$

Given some constant $a\in\mathbb{R}-\{0\}$, find $x_0$ such that $f'(x_0)=0$ where $$f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x.$$ I have managed to write $f'(x_0)=0$ as $$\...
8
votes
1answer
1k views

What is the value of this double sum in closed form?

I encountered the following double sum which requires an evaluation. Is there a closed form for this? $$\sum_{n=0}^{\infty}\frac{\sum_{k=0}^n\binom{n}k^{-1}}{(n+1)(n+2)}.$$ Incidentally, it ...
1
vote
1answer
271 views

Closed form of :$\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ for $ k$ is even integer and :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$

This question is related to my question here such that i want to find a closed form of $\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ , for $k$ is even integer because for odd integer is $0$ as ...
2
votes
0answers
171 views

Is there any precedent in mathematics where closed-form relations between trigonometric and inverse trigonometric functions arise?

This question is connected to my current research where unexpectedly there arise connections between trigonometric/hyperbolic functions and their inverses. In short, if we introduce some element $\...
1
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0answers
57 views

Vibration of point load on a halfspace

The amplitude of vibration of surface of halfspace at a distance r from a point harmonic load of amplitude Q is given by $ w(r,0) = $ $ Q\over 2\pi G $ $ \int_0^\infty $ $ k^{2}\alpha pJ_0(pr)dp \...
-3
votes
1answer
110 views

How to find closed form expression for $\int^\infty_0 \frac{Ar}{1+Cr^\alpha} e^{-Br^2} dr$?

I am badly stuck in some integration here and will appreciate any help out of it. $$\int^\infty_0f(r) dr = \int^\infty_0 \frac{Ar}{1+Cr^\alpha} e^{-Br^2} dr$$ If I let $u = Br^2$, then I get $$ = \...
5
votes
2answers
296 views

What are the most general methods for solving equations in closed form with Lambert W?

What are the most general methods for solving equations with help of Lambert W function or with a generalization of Lambert W function in closed form? I gave a method in MSE here. Which algorithms ...
3
votes
1answer
132 views

Does a linear Recurrence relation with an nonlinear relation between elements can be solved by a closed formula?

I came across with this cool recurrence relation, and unfortunately i couldn't find sufficient mathematical tools to form it to a closed formula. i read several posts from math overflow saying that ...
3
votes
1answer
826 views

Solution of multivariate Geometric Brownian Motion?

It is known how to solve the SDE $dX=X\,dW$ to get a closed form expression of $X(t)=\exp(W_t-\frac{t}{2})$. The question is, is there also a way to solve \begin{equation} \begin{cases} dX=X \, dW_1+...