# Questions tagged [closed-form-expressions]

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

114
questions

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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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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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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**

vote

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

vote

**2**answers

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

**1**answer

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**2**answers

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

**2**answers

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**

votes

**0**answers

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**

votes

**0**answers

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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**0**answers

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**

votes

**0**answers

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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**0**answers

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

**1**answer

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**

vote

**2**answers

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

**1**answer

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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**0**answers

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**

votes

**0**answers

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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**1**answer

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**

votes

**0**answers

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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**0**answers

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**

votes

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**2**answers

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

**0**answers

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**

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**4**answers

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

**1**answer

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

**2**answers

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**

vote

**0**answers

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

**1**answer

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

**2**answers

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

**4**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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**

vote

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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+...