Questions tagged [closed-form]

A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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30 views

Iterated integral involving polylogarithms

To establish notation the polylogarithm Li$_n(x)$ has the power series expansion $$ \text{Li}_n(x)= \sum_{k=1}^\infty \frac{x^k}{k^n} $$ and the Riemann zeta can be considered the special value $\zeta(...
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40 views

Is it possible to simplify $_2F_1(x,-x,x+1,z)$?

Specifically, I'm trying to simplify $_2F_1(1/n,-1/n,1+1/n,z)$. I see that $_2F_1(a+1,b,a,z)$ and $_2F_1(a,b,a,z)$ are better, but I get a zero coefficient when I try to use the standard contiguous ...
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1answer
67 views

Let $F:M\rightarrow N$ and $G:N\rightarrow P$ be surface transformation. Show that $(G\circ F)^{*}=F^{*}\circ G^{*}$

$Let F:M\rightarrow N$ and $G:N\rightarrow P$ be surface transformation. Show that $$(G\circ F)^{*}=F^{*}\circ G^{*}$$ Here is definition : Let $F:M\rightarrow N$ transformation of surfaces. $i$) If $...
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1answer
31 views

Formula for one element in the intersection of two cosets $m\Bbb{Z} + a$ and $n \Bbb{Z} + b$ whenever $\gcd(m,n) = 1$ is true. [duplicate]

We know that $(m\Bbb{Z} + a) \cap (n \Bbb{Z} + b) = \text{lcm}(m,n) \Bbb{Z} + x$, that is if the intersection is not empty, where $x$ is any element of the latter coset. Is there a formula for $x$ in ...
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18 views

L21 norm closed form

Please help me Let X, A, B, and C are 3rd order tensor and a and b are constant. I need the closed form for this problem min_A a*||A||_2,1 +b/2* ||A-(X-B+C/b)||_F^2
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14 views

How to solve exposure gain compensation equation?

I am working on panorama stitching and more specifically exposure compensation between images. I am trying to understand how to solve equation (29) from Automatic Panoramic Image Stitching using ...
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27 views

How to solve a recursive equation with fraction structure

Consider the following recursive equation: $$a(K) = K \frac{a(K-1) + b}{a(K-1) + b + S},K=1, 2, \cdots $$ where $b \in \{1, 2, \cdots \}$ and $S>0$ are both constants. Is there any way to solve ...
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1answer
45 views

Explicit formula for Laurent series of $z^u$ given any complex exponent $u$.

Given any fixed $u\in\mathbb{C}$, does there exist a general form of the $n$-th term of the Laurent series of $z^u$ ? For such $u$ we have $$c_n=\frac{1}{2\pi i}\oint z^u\frac{dz}{z^{1+n}}$$ and $$z^u=...
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22 views

On the angle matching problem, involving inverse $\tan$ function

Given $\mu > 0$ and $X(y)\geq 0$ for all $y \in \mathbb{R}$, let us define \begin{equation} G(X(y)) = \frac{jX(y)+\mu}{jX(y)-\mu} \end{equation} I want to find an expression for $X(y)$ in terms of ...
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2answers
37 views

Any closed form geometric relationships from the data points to the control points of a Bézier curve?

For a quadratic Bézier curve ($b_2(t)$, with control points $p_1, c_2, p_3$), when $t = 0.5$, the interpolant $b_2(t = 0.5)$ is equidistant from $c_2$ and the midpoint $m_2$ of the line from $p_1$ to $...
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1answer
74 views

Proof about the power series of reciprocal multifactorials $m_x(k)=\sum_{n=0}^\infty \frac{x^n}{n\underbrace{!\cdots!}_{\text{k times}}}$

The proof I've attempted mimics very closely the answer on this question. How to prove the formula for the Reciprocal Multifactorial constant? Pre-requisite definitions: A multifactorial of order $k \...
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1answer
108 views

How to estimate the following integral involving exponentials and Bessel functions

Consider the probability distribution $$Q(x)=\frac{\beta x}{2}e^{-\frac{\beta}{4}(a^{2}+x^{2})}I_{0}(\frac{\beta a x}{2})$$ where $a$ and $\beta$ are positive numbers and $I_{0}$ is the modified ...
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1answer
68 views

Closed-form expression for a recursive formula

I'm trying to calculate the closed-form expression of the recursive $a_n = \frac{a_{n-1} + a_{n-2}}{3}$ Where $a_0 = 0, $ $a_1 = \frac {1}{3}$, and $a_2 = \frac {1}{9}$ I have tried to mimic the ...
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346 views

Evaluate $\sum\limits_{n=1}^{\infty}\frac{1}{n^3}\binom{2n}{n}^{-1}$.

Evaluate $$\sum\limits_{n=1}^{\infty}\frac{1}{n^3}\binom{2n}{n}^{-1}.$$ My work so far and background to the problem. This question was inspired by the second page of this paper. The author of the ...
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21 views

Maximum likelihood of a no-closed form density

Supposed we have density function $f(y_i)$ which have no closed form expression. Now can we find the estimating equation of the log likelihood of $f(y_i)$ by taking the derivative(I have tried this ...
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15 views

Simplified expression of no closed-form expression of a density

Suppose $\textbf U\sim N(\textbf0,\textbf I_p )$ and $\textbf {y|u} \sim N(X(t),\sigma_e^2\textbf I_{m})$, Both are basically multivariate Gaussian. What I want is to find the marginal distribution of ...
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34 views

Closed forms for these Fourier series?

I recently encountered the following series $$ \sum_{m \ne 0, m \in \mathbb{Z}} \frac{1}{(\sin m \pi \tau)^n} e^{2\pi i m z} \ , \quad n= 1, 2, ... $$ For $n = 1, 2$, I managed to find the closed ...
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1answer
19 views

Closed-form to this Combinatory problem

Alice sends binary messages to Bob. Every message must end in 1, and the number bits equal to 1 in the message, including the last, cannot exceed a certain $k$ limit. For $n, k ∈ Z> 0$, let $M (n, ...
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14 views

How to plot an equation with no analytical solution (no closed form solutions)

Consider the following equation: $$y-\ln{y}=F(x,k) \ \ \ \ \ \ x,y,k \in \mathbb{R}$$ where $F$ is a non linear real function of $x$ and a constant $k$, that we know. Suppose that we want to use ...
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1answer
94 views

Is $\prod_{n=0}^\infty \left(1-\frac{1}{\cosh ^2((n+1/2)\pi)}\right)=\frac{1}{\sqrt[4]{2}}$ true?

The infinite product $$\prod_{n=0}^\infty \left(1-\frac{1}{\cosh  ^2((n+1/2)\pi)}\right)$$ agrees with $\frac{1}{\sqrt[4]{2}}$ to at least 100 decimal places. The "identity" is reminiscent ...
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2answers
39 views

How to derive the closed-form formula for the square sum through integration?

I know that the closed-form of $\sum_{i=1}^n i^2$ is $n(n+1)(2n+1)/6$. But how to derive the formula through integration? My attempt is $$ \sum_{i=1}^n i = \frac{n^2+n}{2}\\ \frac{1}{2}\sum_{i=1}^n i^...
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4answers
100 views

Ideas for a Closed form for $ \sum_{k=0}^n k10^k$

Is there a closed formula for this summation: $$ \sum_{k=0}^n k10^k, $$ where $n\in\mathbb{N}$? I would like to learn trick o strategies for this kind of problems.
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1answer
65 views

What general kinds of closed-form problems are there?

A closed-form expression is a mathematical expression that contains only finite numbers of symbols and operations from a given set. A mathematical problem is a closed-form problem if its solution is ...
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1answer
29 views

Understanding Wilfian formula?

In this paper by Igor Pak, he mentions the following definition of wilfian formulas, which basically aim at characterising what amounts to a good enumeration formula, the classification goes as ...
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1answer
266 views

About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-_{3}F_{2}\left(\frac{1}{4},\frac{1}{4},\frac{1}{2};\frac{5}{4},\frac{5}{4};-1\right).\...
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176 views

Challenging integral $I=\int_0^{\pi/2}x^2\frac{\ln(\sin x)}{\cos x}dx$

My friend offered to solve this integral. $$I=\int_0^{\pi/2}x^2\frac{\ln(\sin x)}{\cos x}dx=\frac{\pi^4}{32}-{4G^2} $$ Where G is the Catalan's constant. $$I=\int _0^{\infty }\frac{\arctan ^2\left(u\...
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11 views

Integer Valued Lexicographically Ordered Matrices

I am interested in getting a better understanding of the matrices in the following set $$X_{m,n,k} = \left\{A =[a_{i,j}] \in \mathbf{N}^{m,n}: \text{ the rows of $A$ are lexicographically ordered and $...
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2answers
117 views

Evaluating $\int_{0}^{\infty} \left( \text{coth} (x) - x \text{csch}^2 (x) \right) \left( \ln \left( \frac{4 \pi^2}{x^2} + 1 \right) \right) \, dx$

How can the following improper integral be evaluated? $$\int_{0}^{\infty} \left( \text{coth} (x) - x \text{csch}^2 (x) \right) \left( \ln \left( \frac{4 \pi^2}{x^2} + 1 \right) \right) \, dx$$ or ...
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0answers
27 views

Is it possible to find the closed-form solution of the following problem?

I was stuck for a long time to solve the following problem \begin{equation}\label{R_fixed_rho} \mathop {\max }\limits_{\{0 \leq r_t \leq C \}} \left\{ \mathop {\min }\limits_{ {\cal S} \subseteq {\cal ...
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7answers
2k views

How to evaluate $\sum_{k=3}^{\infty} \frac{\ln (k)}{k^2 - 4}$?

Is it possible to evaluate the sum: $$\sum_{k=3}^{\infty} \frac{\ln (k)}{k^2 - 4}$$ I expect it may be related to $\zeta^{\prime} (2)$: $$\zeta^{\prime} (2) = - \sum_{k=2}^{\infty} \frac{\ln(k)}{k^2}$$...
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2answers
55 views

Is there another formula which generates Pythagoras' triples such that the largest $2$ of the triple differ by $3$?

I was thinking about Pythagoras' triples recently, and I wondered if I could find a formula that generated Pythagorean triples such that the largest of $2$ numbers of the triple differ by $1$, and I ...
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0answers
37 views

What can we say about the closed-form solutions of $ x^q - 2x + 1$ for $\def\N{\mathbb{N}} q \in \N^+$?

$\def\N{\mathbb{N}}$ What can we say about the closed-form solutions for $x^q - 2x + 1$ for $q \in \N^+$ ? I can solve it for small values of $q$ up to 5. And I wonder what happens next, for $q > 5$...
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2answers
110 views

Does $\int_1^{\infty}\frac{x^2\tan^{-1}(ax)}{x^4+x^2+1}dx$ have closed form?

I have been trying to find the closed form for integral below $$\int_1^{\infty}\frac{x^2\tan^{-1}(ax)}{x^4+x^2+1}dx ,\; \; a>0 $$ My progress to this integral $$\cong\frac{\pi^2}{8\sqrt 3}+\frac{\...
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0answers
17 views

How to express $n$ unknown variables in a system with $n-1$ equations

I have a system of $n-1$ equations which have $n$ unknown variables, as follows: \begin{align} &\text{Equation $1$} &&x_1A_{1,1}+...+x_nA_{1,n}=0\\ & \hspace{1cm}\vdots && \...
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0answers
71 views

Closed form of $\int_0^{\pi /2} \operatorname{F}(x,i)\sin nx\, \mathrm dx$ for odd $n$

Let $\operatorname{F}$ be the incomplete elliptic integral of the first kind with modulus $i$ (the imaginary unit). Then (see Integrals and Series Vol. 3 by Prudnikov et al., p. 34): $$\begin{align}\...
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1answer
27 views

Closed form for this power series looking like an hypergeometric?

I would like to "resum" the following expression: $$\sum_{k=0}^a \frac{(-a)_k (-b)_k}{(c)_k} x^k\,, \tag{1}$$ with $a, b, c$ positive even numbers and $x > 0$ real. Is there a known ...
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3answers
77 views

How to solve $\cos x = x \sin x$

Is it possible to express the solutions to $\cos x = x \sin x$ in closed form? Numerically, the first positive solution seems to be $x = 0.8603335890193...$, which is is suspiciously close to $\frac{\...
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0answers
26 views

Closed form of some interesting indefinite integrals

While studying analytic number theory, the following indefinite integrals fascinated me the most: $$\int_{0}^{\infty}\dfrac{\cos(nx)}{(x^2 + 1)^{z + 1}}\log x\,dx + \dfrac{\pi}{2}\int_{0}^{\infty}\...
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2answers
47 views

Computing $\int_0^1 \frac{(1-t)(1+at)}{|1+at|^p} dt$.

Just as the title suggests, I want to compute the integral of the form $$ \int_0^1 \frac{(1-t)(1+at)}{|1+at|^p} dt $$ in terms of $a$ and $p$, where $a \in \Bbb R$ and $p\in (1,2)$. This is not a ...
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2answers
32 views

Inverse is not connected

Let $f:[a, b] \rightarrow[a, b]$ be continuous. Provide a counterexample for the following statement: The inverse image $f^{-1}([c, d])$ is connected for any $[c, d] \subset[a, b]$. One of the ideas ...
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0answers
21 views

Incomplete upper gamma for a non-integer number of degrees of freedom

I can't seem to nail the closed form of the incomplete upper gamma function for the number of degrees of freedom $s$ being a fraction type $n/2$ where $n$ is integer. For the case when $s$ is integer ...
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0answers
67 views

Closed form of $\sum_{n=1}^{\infty} \frac{a^n}{n(n+1)}$

I have come across the series$$\sum_{n=1}^{\infty} \frac{a^n}{n(n+1)}, \quad|a|<1$$ in an intermediate step while solving a problem. I wonder if there is a closed form for it? I could only observe ...
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1answer
168 views

Prove $\int_{0}^{\pi}\frac{u}{1-\cos u}\ln\frac{1+\sin u}{1-\sin u}{d}u=\left(\pi+2\ln2\right)\pi$ [closed]

How to prove $$\displaystyle\int_{0}^{\pi}\frac{u}{1-\cos u}\ln\left(\frac{1+\sin u}{1-\sin u}\right)\mathrm{d}u=\left(\pi+2\ln2\right)\pi\,\,?$$ I tried to apply the Feynman method to get the ...
1
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1answer
71 views

Closed form expression for eigenvectors of 2x2 matrix?

Is there a closed form expression for the eigenvalues/eigenvectors of an arbitrary 2x2 matrix $ \begin{bmatrix} a & b \\ c & d \end{bmatrix} $? Wolfram|Alpha tries to provide an ...
2
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0answers
37 views

Closed form for a minimum related to Khintchine inequality

Let $p>2$ be a real number. In this blog post by George Lowther, a proof of the right-hand Khintchine inequality is given where $$m_p:=\min_{x>0}\;x^{-p}\cosh(x)$$ comes into play. Question: Is ...
2
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1answer
56 views

Closed form for definite integral involving Bessel function, $K_1$

Wolfram Alpha knows that, and it can be calculated that: $$ \int_0^1 \exp\bigg(\frac{1}{\log x}\bigg)~dx=2K_1(2).$$ Where $K_1$ is a modified Bessel function of the second kind. I wanted to find out ...
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1answer
39 views

Finding open / close form from this recurrence relation [closed]

In Nlogonia, buses and minibuses are 18 and 12 meters long, respectively. Buses can only be green; the minibuses can be blue or red. Two vehicles of the same color are indistinguishable. One of the ...
3
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0answers
93 views

Is there a closed form of $\sum_{k=1}^{n} \frac{|S_{1}(n,k)|}{k!}z^{k} $?

So far, I have found (p. 5) the following generating functions of the unsigned Stirling numbers of the first kind: $$ \sum_{l=1}^{n} |S_{1}(n,l)|z^{l} = (z)_{n} = \prod_{k=0}^{n-1} (z+k) , $$ and $$ \...
3
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1answer
101 views

iterated function $a(n) = \lfloor n\phi + 0.5\rfloor$

Let $a(n) = \lfloor n\phi + 0.5\rfloor$ for all $n \geqslant 1$, where $\phi$ is the golden ratio. Now let $$a(n)^k = a(a(\ldots(a(n))))$$ where we have iterated $a(n)$, $k$ times in the RHS. I am ...
2
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0answers
22 views

Summation with given parameter :

Consider the following sum: $$F(x)=\sum_{n=2}^{\infty}\frac{1}{(n\ln(n))^x}$$ Now this series converges for $x>1$. Can we get a closed form of this function for $x>1$?

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