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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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impossibility of closed form solution to ∫ sin(sin(x)) dx

I've reviewed the general theory at How can you prove that a function has no closed form integral? and the links therein. But they do not give me sufficient traction to show the impossibility of ...
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1answer
26 views

Closed-form solution for $f(x)/x=y$ using $f^{-1}$

I'm programming a piece of math that requires solving an equation of a form $f(x)/x=y$. Now I already have $f^{-1}(z)$ coded (efficiently, and not by me) so I'd prefer using this implementation ...
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0answers
42 views

$ t(n) = t( x_1 x_2 x_3 …) = t(x_1) + t(x_2) + t(x_3) + … + t( x_1 + x_2 + x_3 + … ) $

Let $ n > 1 $ be an integer. Consider The prime factorization $$ n = x_1 x_2 x_3 ... $$ Now define $$ t(n) = t( x_1 x_2 x_3 ...) = t(x_1) + t(x_2) + t(x_3) + ... + t( x_1 + x_2 + x_3 + ... ) $$...
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Conjecture: $\sum\limits_{n\geq0}\left(\frac12\right)^n\prod\limits_{k=1}^{n}\frac{2n-2k+1}{2n-2k+2}=\sqrt2$ [duplicate]

I am trying to solve a problem I made form myself: proving that $$\sum_{n\geq0}\left(\frac12\right)^n\prod_{k=1}^{n}\frac{2n-2k+1}{2n-2k+2}=\sqrt2$$ The highly accurate powers of Desmos seem to ...
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30 views

Does it exist a closed form for $\sum_{i=1}^{n}\frac{i!}{x^i}$?

From a combinatorial problem, I found the truncated sum $$f(n) = \sum_{i=1}^{n}\frac{i!}{x^i}.$$ I wonder if there is any compact way of rewriting $f(n)$. In my case $x = 1/2$, but the general case ...
1
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1answer
19 views

How to find the closed form of the recursive equation, if 2 of the 3 roots of the characteristic equation are imaginary and only 1 root is real?

The given recursive equation is $$f(n) = f(n-1) + 3f(n-3) + 2n$$ The characteristic equation for $$f(n) = f(n-1) - 3f(n-3)$$ is $r^3 - r^2 - 3 = 0$, which has $2$ imaginary roots and $1$ real root ...
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1answer
30 views

Is there a closed-form formula for the derivative of the orthogonal polar factor of a matrix?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\newcommand{\SO}{\operatorname{SO}_n}$ $\...
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2answers
164 views

Closed Form of the Real Portion of $f(n) = \prod_{m=2}^{n-1} e^{\pi i n/m}$

I am wondering if it is possible to express an equation in closed form. I currently have: $$f(n) = \prod_{m=2}^{n-1} e^{\pi i n/m}$$ Where $i$ is the $\sqrt{-1}$, which I know it commonly ...
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3answers
240 views

Finding a closed form for $\int_{0}^{1}\frac{\ln\left ( 1-x^{2} \right )\arcsin ^{2}x}{x^{2}}\mathrm{d}x\approx -0.939332$ [duplicate]

I'm attempting to find a closed form for $$\int_{0}^{1}\frac{\ln\left ( 1-x^{2} \right )\arcsin ^{2}x}{x^{2}}\mathrm{d}x\approx -0.939332$$ I tried to use $$\displaystyle \arcsin^{2}x=\frac{1}{2}\...
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2answers
43 views

Evaluating Dirichlet $L$-functions at $s=1$

I'm trying to find references on general methods for evaluating Dirichlet $L$-functions at $s=1$, but it's proving a little harder to google than I'd hoped. Specifically I'm looking for any books or ...
3
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2answers
142 views

Studying on this sum interesting $\sum_{n=1}^{\infty}\frac{{2n \choose n }}{4^n n}$

I was studying this particular sum $$\sum_{n=1}^{\infty}\frac{{2n \choose n}}{4^n n}$$ and eventually I ended up with is sum $(1)$ $$\sum_{n=1}^{\infty}\frac{{2n \choose n}}{(-4\phi)^n}\cdot\frac{1}{(...
3
votes
3answers
48 views

Find closed form of sum of fraction of binomial coefficients

can somebody give me a hint for this exercise, where I have to find the specific closed form? $\sum_{k=0}^m \frac{\binom{m}{k}}{\binom{n}{k}}, m,n\in\mathbb{N}$ and $m\leq n$ What I have done so ...
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2answers
375 views

Integral $\int_0^\frac{\pi}{2} x^2\sqrt{\tan x}\,\mathrm dx$

Last year I wondered about this integral:$$\int_0^\frac{\pi}{2} x^2\sqrt{\tan x}\,\mathrm dx$$ That is because it looks very similar to this integral and this one. Surprisingly the result is quite ...
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0answers
43 views

Sum with harmonic number of squared argument as e.g. $\sum_{k=1}^\infty \frac{H(k^2)}{k^2}$

I wonder if closed expression can be found for sums of harmonic numbers with a squared argument. Examples are $$s_{1}=\sum_{k=1}^\infty \frac{ H(k^2)}{k^2} \simeq 3.28709\tag{1}$$ $$s_{2}=\sum_{k=1}^...
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2answers
156 views

Closed form for $\sum\limits_{n=2}^{\infty}\frac1{n^3-1}$

I am investigating (just for fun) the sum $$S=\sum_{n=2}^{\infty}\frac1{n^3-1}$$ Wolfram Alpha gives me the 'value' $$S=-\frac13\sum_{\{\omega\,:\,\omega^3+6\omega^2+12\omega+7=0\}}\frac{\psi_{0}(-\...
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+50

What is the surface area of the 3-dimensional elliptope?

The $n$-elliptope is defined as the set of $n$-by-$n$ correlation matrices; that is, the set of $n$-by-$n$ symmetric positive-definite matrices with ones on the diagonal. Such matrices are ...
3
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3answers
99 views

Closed form of $\int_0^\infty \sin(x)\sin\left(\frac{1}{x}\right)dx$?

I have stumbled onto an interesting integral$$\int_0^\infty \sin(x)\sin\left(\frac{1}{x}\right)dx$$ which I noticed graphically that it appears to be $1$, but I have no idea on how to evaluate it. ...
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2answers
33 views

Is every sequence bounded by a sequence which can be represented in closed form?

Let $X$ be the set of $\mathbb{R}$-valued sequences, i.e. $X := \mathbb{R}^{\mathbb{N}}=\{f: \mathbb{N} \to \mathbb{R}\}$, and let $S$ the set of sequences which can be expressed in closed form, i.e.: ...
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2answers
59 views

For the periodic sequence, is there always an algebraic closed form?

This question is a generalized form of the problem I asked before: Algebraic Closed Form for $\sum_{n=1}^{k}\left( n- 3 \lfloor \frac{n-1}{3} \rfloor\right)$ Let, look at this periodic sequence: ...
7
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1answer
215 views

Infinite summation formula for modified Bessel functions of first kind

I was trying to find a closed form for the integral $$4\int_0^{\pi/2} t \, I_0(2\kappa\cos{t}) dt \; ,$$ where $$I_{\alpha}(z) := i^{-\alpha}J_{\alpha}(iz) = \sum_{m=0}^{\infty}\frac{\left(\frac{z}{...
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5answers
73 views

Algebraic Closed Form for $\sum_{n=1}^{k}\left( n- 3 \lfloor \frac{n-1}{3} \rfloor\right)$

Let's look at the following sequence: $a_n=\left\{1,2,3,1,2,3,1,2,3,1,2,3,...\right\}$ I'm trying to calculate: $$\sum_{n=1}^{k} a_n$$ Attempts: I have a Closed Form for this sequence. $$...
1
vote
1answer
56 views

Every differential form $\omega$ of degree 1 in the sphere $S^m\subset\mathbb{R}^{m+1}$ that is closed is also exact.

I am in a series of self-studies on differential forms on $m$-dimensional surfaces in Euclidean space. I'm two days thinking about the exercise below. The book that proposes this exercise gives no ...
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2answers
39 views

Does this bivariate function have a non-summation form or a good looking generating function?

The function is this one: For $m,n$ positive integers, $$a(m,n)=\sum_{k=0}^n\sum_{i=0}^m {n\choose k}{m\choose i}(-1)^{n-k}(-1)^{m-i}2^{ki}$$ $$ = \sum_{k=0}^n {n\choose k}(-1)^{n-k}(2^k-1)^m.$$ I ...
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0answers
27 views

Can this Helmholtz PDE with Robin boundary conditions be solved analytically?

Consider the following Helmholtz problem in the infinite triangle $y>0,\;x>y$ with parameters $Q<0$, $P\ge0$, $P<|Q|$. $$\left\{\begin{align} &\psi^{(2,0)}(x,y)+\psi^{(0,2)}(x,y)+E\...
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0answers
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Determine $\kappa$ / does there exist a closed form solution?

How can I determine the value for $\kappa$ for e.g. $n \in \{200, 400, 600\}$. Is there a closed form solution? Let $a_j = j^{0.51}$ and let $\kappa$ be the solution to the equation $$ \sigma^2_{\...
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0answers
95 views

Scary looking integral from a movie background

I watched recently this movie: https://www.imdb.com/title/tt3149038/mediaviewer/rm261224704 and saw on the poster background, (top-left) the following integral $$\int_{\large\frac{v}{\sqrt{t}}}^{+\...
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0answers
32 views

Possible closed form approximation of a trigonometrical expression

I need to create a simple algorithm to draw a Venn diagram (ideally for 3-circle case, but even solving it for 2 is a good start). So given thee numbers - X & <...
2
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3answers
128 views

Prove that $ \int\limits_0^\infty\frac{x^2}{e^{2 x} (1-x)^2+e^{-2 x} (1+x)^2}\,dx=\dfrac{3\pi}{8} $ [closed]

Is this conjecture true? Conjecture: $$ \int_0^\infty\frac{x^2}{e^{2 x} (1-x)^2+e^{-2 x} (1+x)^2}\,dx=\dfrac{3\pi}{8} $$ I found it myself based on numerical evidence. Need help in analytical ...
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0answers
28 views

Solving linear ordinary, 2nd order differential equations via global integral bases.

Consider a linear, homogenous 2nd order ODE: \begin{equation} L\left[y(x)\right] = \left[\frac{d^2}{d x^2} + a_1(x) \frac{d}{d x} + a_0(x)\right] y(x)=0 \end{equation} In https://arxiv.org/pdf/1606....
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1answer
47 views

Closed form for $\sum\limits_{n=1}^{\infty}\frac{O_{n}^{(p)}}{(2n-1)^{q}}$, with $O_{n}^{(s)}=\sum\limits_{k=1}^n\frac1{(2k-1)^{s}}$

Consider the sum $$\sum_{n=1}^{\infty}\frac{O_{n}^{(p)}}{(2n-1)^{q}}\text{, with }O_{n}^{(s)}=1+\frac{1}{3^{s}}+\dots+\frac{1}{(2n-1)^{s}}$$ My question is: if there exists some general theorems ...
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0answers
14 views

Double sum over $\left\lfloor{ac+bd\over k}\right\rfloor$

We have $$\left\lfloor{ac+bd\over k}\right\rfloor-\left\lfloor{ac+bd-1\over k}\right\rfloor=1-\left\lceil{ (ac+bd)\mod{k}\over k}\right\rceil$$ for $a,b,c,d,k$ - integers, $a\geqslant0$, $b\geqslant0$,...
6
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1answer
116 views

Generalizing $\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)}=2$

I was looking at this paper on section [17], $$\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)}=2\tag1$$ Let generalize $(1)$ $$\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)(...
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2answers
91 views

Closed expression for alternating harmonically wrapped harmonic series $\sum_{n=1}^\infty (-1)^{n+1} H_{\frac{1}{n}}$

It is well known that the alternating harmonic sum $\sum_{n=1}^\infty (-1)^{n+1} \frac{1}{n}$ converges to $\log(2)$. Now let us wrap $\frac{1}{n}$ with the harmonic number $H_k$ (continued ...
3
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1answer
130 views

Is the closed-form of $\sum_{k=1}^{\infty}\frac{e^{-k}}{k^2}$ known?

I'm looking for an expression that yields the above sum. It's a straight and simple question, feel free to move this post to any other section, if it's not in the appropriate section, I know this ...
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3answers
60 views

How to prove that $\sum_{n=k}^{\infty}{n-1 \choose k-1}x^{n}=\left(\frac{x}{1-x}\right)^{k}$? [closed]

Im looking to show that $$\sum_{n=k}^{\infty}{n-1 \choose k-1}x^{n}=\left(\frac{x}{1-x}\right)^{k}$$ and i dont really know how to do it. Any suggestions?
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3answers
69 views

$F_k(a_1,b_1,c_1;a_2,b_2,c_2)=\int_0^k\frac{a_1x^2+b_1x+c_1}{a_2x^2+b_2x+c_2}\mathrm{d}x$

I would like to know a general closed form for $$J=F_k(a_1,b_1,c_1;a_2,b_2,c_2)=\int_0^k\frac{a_1x^2+b_1x+c_1}{a_2x^2+b_2x+c_2}\mathrm{d}x$$ Where $k>0$, $4a_1c_1-b_1^2<0$, and $4a_2c_2-b_2^2<...
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2answers
124 views

Sum of $\sum_{n=1}^\infty\frac{\ln(n) -\ln(n+1)}{n+1}$

The sum series exercise started as: $$\sum_{n=1}^\infty\frac{\ln\left(\frac{n^n}{\left(n+1\right)^n}\right)}{n(n+1)} = \sum_{n=1}^\infty \frac{n\ln\frac{n}{n+1}}{n(n+1)} = \sum_{n=1}^\infty \frac{\ln(...
5
votes
4answers
199 views

Evaluate $\int_0^1 \frac{\operatorname{arctanh}^3(x)}{x}dx$

I'm trying to evaluate the following integral: $$\int_0^1 \frac{\operatorname{arctanh}^3(x)}{x}dx$$ I was playing around trying to numerically approximate the answer with known constants and found ...
1
vote
1answer
77 views

Is there any solution (numeric or closed) to integration of ${\sin x}^{\cos x}$?

I've tried so many ways to evaluate $\int{\sin x}^{\cos x}dx$ and even searched and used programs like matlab, maple and scipy library and got no answer! my question is clear, is there any numerical ...
1
vote
2answers
48 views

Number of possible bishop moves on an $n \times m$ chessboard

For rook we have obviously $$R(n,m)=nm(n+m-2)$$ and for bishop $$B(n,m)=4\left(m\binom{n}{2}-\binom{n+1}{3}+\binom{n-m+1}{3}\right)$$ if we assume $\binom{n}{k}=0$ for $n<0$. Is there a way to ...
0
votes
2answers
82 views

A Closed form for the $\sum_{n=1}^{\infty}\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3}$

I'm looking for a closed form for this sequence, $$\sum_{n=1}^{\infty}\left(\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3} \right)$$ I applied convergence test. The series converges.I want to know ...
4
votes
2answers
114 views

Closed form of this type $\sum_{j=0}^{\infty}\frac{2^jj^n}{(2j+1)(2j+3){2j \choose j}}$

Given that, $$\sum_{j=0}^{\infty}\frac{2^j\left(j-\frac{1}{3}\right)^3\left(j^2+j-1\right)}{(2j+1)(2j+3){2j \choose j}}=A\tag1$$ We have $A=2\pi+12+\frac{1}{3}?$ We can generalize the above $(1)$: $...
7
votes
1answer
163 views

Is there a closed form for the integral $\int_0^\infty \frac{e^{-x^2} I_0 \left(\beta x \right) d x}{\sqrt{ \alpha^2+x^2}}$

I encountered this integral in my work, and it would be really convenient if it had a closed form in terms of any known special functions (which Mathematica could handle): $$J(\alpha,\beta)=\int_0^\...
29
votes
2answers
741 views

Proving that $\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$

The following integral was proposed by Cornel Ioan Valean and appeared as Problem $12054$ in the American Mathematical Monthly earlier this year. Prove $$\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{...
5
votes
1answer
84 views

What is the inverse of simply composited elementary functions?

$A$ be an elementary function, algebraic over $\mathbb{C}$, $f_1$ and $f_2$ are bijective elementary functions with elementary inverses, $F\colon z\mapsto A(f_1(f_2(z)))$ be a bijective elementary ...
2
votes
0answers
41 views

Closed form expression for (periodic) generalized harmonic numbers?

As far as I understand, there does not exist a pure closed form expression for the generalized harmonic numbers $H_{n,m}=\sum_{k=1}^n \frac{1}{k^m}$ with $m\in\mathbb R$. My question is, however, ...
9
votes
3answers
112 views

How to prove $\int_0^\infty \ln(1+\frac{z}{\cosh(x)})dx=\frac{\pi^2}{8}+\frac{(\cosh^{-1}(z))^2}{2},z\ge1$ and a closed form for $-1<z<1$?

I observed graphically that $$f(z)=\int_0^\infty \ln\left(1+\frac{z}{\cosh(x)}\right)dx=\frac{\pi^2}{8}+\frac{(\cosh^{-1}(z))^2}{2},z\ge1$$ Can anyone explain why this holds? I tried differentiating ...
11
votes
5answers
331 views

Evaluate $\int_{-\pi/4}^{\pi/4}\frac{x}{\sin x}\mathrm{d}x$

I am working on the integral $$I=\int_{-\pi/4}^{\pi/4}\frac{x}{\sin x}\mathrm{d}x=2\int_0^{\pi/4}\frac{x}{\sin x}\mathrm{d}x$$ Which I am fairly confident has a closed form, as $$\int_{0}^{\pi/2}\...
2
votes
0answers
44 views

Is there a closed-form formula for the derivative of the positive factor of a matrix?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\renewcommand{\skew}{\operatorname{skew}}$ $\...
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votes
0answers
20 views

Inverse of a number-valued function on a curve

$S=\{(x_1,x_2)\ |\ \forall x\in\mathbb{R}\colon x_1=x,x_2=\exp(x)\}$ be a set, and $f$ be the function $f\colon S\to \mathbb{R},(x1,x2)\mapsto x_1+x_2$. How can I show if $f$ is bijective? How can ...