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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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how to integrate $\int_0^1 \ln^4(1+x) \ln(1-x) \, dx$?

I'm trying to evaluate the integral $$\int_0^1 \ln^4(1+x) \ln(1-x) \, dx,$$ and I'd like some help with my approach and figuring out the remaining steps. or is it possible to evaluate $$\int_0^1 \ln^n(...
Martin.s's user avatar
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4 votes
3 answers
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How to Find closed form $\sum_{n=1}^{\infty}\frac{x^{2n}}{n}a_n$

How to Find closed form :$$\sum_{n=1}^{\infty}\frac{x^{2n}}{n}a_n$$ where $a_n=\sum_{k=1}^{2n-1}\frac{(-1)^{k-1}}{k}$ $$S(x)=\sum_{n=1}^{\infty}\frac{x^{2n}}{n}a_n=\sum_{n=1}^{\infty}\int^1_0y^{n-1}x^{...
Mostafa's user avatar
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5 votes
2 answers
142 views

The "seashell constant": closed form for $\frac12\exp\int_0^1-\log(\sin(\frac{\pi}{6}+\frac{2\pi}{3}x))\mathrm dx$?

I am looking for a closed form for $R=\frac12\exp\int_0^1-\log\left(\sin\left(\frac{\pi}{6}+\frac{2\pi}{3}x\right)\right)\mathrm dx\approx0.6159$. Wolfram does not give a closed form for $R$. Wolfram ...
Dan's user avatar
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5 votes
1 answer
111 views

How can I show that $\int_0^{\frac{\pi}{2}}\sin\left(\frac{x}{2}\right) \text{arctanh}\left(\sin(2x)\right) \ dx$.

Question: How can I show that $$\int_0^{\frac{\pi}{2}} \sin\left(\frac{x}{2}\right) \text{arctanh}\left(\sin(2x)\right) \ dx=\log\left(\left(2\sqrt{2-\sqrt{2}}+2\sqrt{2}-1\right)^{\sqrt{2+\sqrt{2}}} \...
Martin.s's user avatar
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Is there a closed form method of expressing the *content* of integer partitions of $n$?

I know that the question of a closed form for the number of partitions of $n$, often written $p(n)$, is an open one (perhaps answered by the paper referred to in this question's answer, although I'm ...
julianiacoponi's user avatar
1 vote
0 answers
49 views

Solution of Sturm-Liouville Problem with variable coefficients

I am reading a paper and the authors solve the Sturm-Liouville equation $$(a^2-1)Q''(a) + \left(\frac{8}{3}a - \frac{2}{a}\right)Q'(a) + \frac{4}{9}Q(a)=0$$ on $0\le a<1/2$. They claim that the two ...
Diffusion's user avatar
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7 votes
2 answers
100 views

How to Evaluate the Integral $\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{1+\sin(y)}\ln(\sin(y))}{\cos(y)}dy?$

Question: How to Evaluate the Integral $$\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{1+\sin(y)}\ln(\sin(y))}{\cos(y)}dy?$$ My attempt I'm looking for a method to evaluate it. I've attempted a substitution to ...
Martin.s's user avatar
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4 votes
1 answer
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how to evaluate $\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{\sqrt{xy}(1+xy) \log_{\pi}^{2}{xy}} \, dx \, dy$

How to evaluate: \begin{align*} &\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{(\sqrt{xy}+\sqrt{x^{3} y^{3}}) \left[ \log_{\pi}^{2}{x} + 4\log_{\pi}{\sqrt{x}} \log_{\pi}{y} + \log_{\pi}^{2}{y} \right]} \, ...
Martin.s's user avatar
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2 votes
2 answers
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A question about sum with reciprocal quartic

Evaluate $$ \sum_{n=1}^{\infty} \frac{n+8}{n^{4}+4} $$ According to WolframAlpha it is $\pi\coth(\pi) - \dfrac{5}{8}$ My attempt: I tried to separate $\dfrac{n}{n^{4}+4}$ and $\dfrac{8}{n^{4}+4}$. ...
Briston's user avatar
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3 votes
1 answer
123 views

how to evaluate this integral $\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$

Question statement: how to evaluate this integral $$\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$$ I don't know if there is a closed form for this integral or not. Here is my attempt to solve ...
Martin.s's user avatar
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1 vote
1 answer
134 views

Evaluate $\lim_{n\to\infty}\prod_{k=1}^n \frac{2n}{e}(\arcsin(\frac{k}{n})-\arcsin(\frac{k-1}{n}))$

I'm trying to evaluate $L=\lim\limits_{n\to\infty}f(n)$ where $$f(n)=\prod\limits_{k=1}^n \frac{2n}{e}\left(\arcsin\left(\frac{k}{n}\right)-\arcsin\left(\frac{k-1}{n}\right)\right)$$ We have: $f(1)\...
Dan's user avatar
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7 votes
3 answers
232 views
+50

how to evaluate $\int_0^1{\ln ^3\left( 1-x \right) \ln ^2\left( 1+x \right) \text{d}x}$

Integral: how to evaluate $$\int_0^1{\ln ^3\left( 1-x \right) \ln ^2\left( 1+x \right) \text{d}x}$$ Same context I'm not sure of the closed form of the integral, as I haven't evaluated it yet. ...
Martin.s's user avatar
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0 votes
1 answer
30 views

Closed form expression for series $f_\alpha(x) = \sum_{n\geq 0} (-n)^{\alpha} x^{2n}$

Let $\alpha \in \mathbb N$, and consider the Taylor series $$f_\alpha(x) = \sum_{n >0} (-n)^{\alpha} x^{2n}$$ which is convergent for $|x|<1$. Question: can we find a closed form to express $...
Overflowian's user avatar
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1 vote
0 answers
78 views

Closed form for $ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\infty \dfrac1{(a_1!+a_2!+\ldots+a_n!)} $ [closed]

After reading this post and the general solution for that case, I wonder if there is a closed form for the general solution for this sum: $ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\...
user967210's user avatar
3 votes
1 answer
87 views

Least number of circles required to cover a continuous function on a closed interval.

Now asked on MO here. This question is a generalisation of a prior question. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of circles with radius $r$ required to ...
pie's user avatar
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7 votes
3 answers
412 views

How to evaluate $\int_0^1 \ln ^3(1+x) \ln (1-x) d x$?

QUESTION:How to evaluate $$\int_0^1 \ln ^3(1+x) \ln (1-x) d x$$? I'm not sure of the closed form of the integral, as I haven't evaluated it yet. However, after evaluating the integral $$\int_0^1 \ln (...
Martin.s's user avatar
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0 votes
0 answers
39 views

Does this hypergeometric function have a closed-form or special value?

$$_{3}F_{2}(1,1,1;\frac{3}{2},\frac{3}{2};z) $$ I don't know much about hypergeometric functions, and I'm not sure if it has a closed-form or special value.
Loyar's user avatar
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2 votes
0 answers
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Closed form for a differential equation with sin

I am currently trying to model some dynamics for a project of mine. But I got following differential equation: $$\frac{d^2\theta}{dt^2}=a\theta+b\cdot \sin(\theta-c)$$ Where $a,b,c,\theta \in \mathbb{...
Fingeg's user avatar
  • 21
0 votes
1 answer
105 views

Identify the special function with this sum and integral form

There is a multivariate generalization of the Bessel function that has both a sum and integral form. Both are functions of a vector $\mathbf{0}\leq\mathbf{x}\in\mathbb{R}^n$, with parameters defined ...
Victor V Albert's user avatar
0 votes
1 answer
65 views

closed form of $\sum_{n=1}^\infty \frac{J_{2n-1}(z)}{2n-1}$

Does the following infinite series have a closed form: \begin{equation} \sum_{n=1}^\infty \frac{J_{2n-1}(z)}{2n-1}? \end{equation} Here, $J_n$ is the Bessel function. (If the denominator does not ...
user1239110's user avatar
1 vote
0 answers
56 views

How to integrate $\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$ [duplicate]

Question; how to integrate $$\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$$ here is my attempt to solve the integral \begin{align} I&=\int_0^\frac{1}{2}\frac{\ln(1+...
Martin.s's user avatar
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2 votes
1 answer
46 views

"Closed form" as a conventional/contextual notion

I have read (Borwein & Crandall, 2013) that the notion of "closed form" (e.g. solution, expression, number) is somehow conventional, that what has been regarded as a closed form solution ...
user avatar
0 votes
0 answers
79 views

Closed-form solution for series $\sum_{n=1}^{\infty}\frac{a^{n}}{\sqrt{n!}}$ involving square root of a factorial

We have $e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}$, however, we are facing $\sum_{n=1}^{\infty}\frac{a^{n}}{\sqrt{n!}}$ involving square root of a factorial. Since $n!>n^{2}$ if $n\geq4$, so $\...
Chen Deng-Ta's user avatar
5 votes
1 answer
224 views

Closed form for $\Gamma(a-x)$ where $a \in (0,1]$.

Now asked on MO here. I wonder if there is a closed form for $ \Gamma(a-x)$. And by closed form here I mean a finite combinations of elementary functions, powers of $\Gamma(a)$ and powers of $\Gamma(...
pie's user avatar
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0 votes
0 answers
30 views

$ \lambda^{*}(n) $ minimal polynomials

I already asked a closely related question on MSE, but didn't received any answer. Considering the modular lambda function, the values of $ \lambda^{*}(n) $ for some integers are given on here. Is ...
user967210's user avatar
2 votes
0 answers
197 views

Showing $\int_{0}^{1}\frac{E(\tfrac{x}{\sqrt{x^2+8}})}{\sqrt{8-7x^2-x^4}}dx=\frac{1}{3}K(\frac{1}{\sqrt{2}})E(\frac{1}{\sqrt{2}})$

Context $\begin{align} K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2t}}\tag{1} \end{align}$ and $\begin{align} E(k)=\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2t}dt\tag{2} \end{align}$ the complete elliptic ...
User's user avatar
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9 votes
1 answer
370 views

evaluation of $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{H_{n} H_{n+1}^{(2)}}{(n+1)^{2}}$ and other Euler sums

I was trying to evaluate this famous integral $$\int_{0}^{1} \frac{\ln (x) \ln^{2}(1+x) \ln(1-x)}{x} \ dx $$ Here is my attempt so solve the integral \begin{align} &\int_{0}^{1} \frac{\ln (x) \ln^{...
user avatar
7 votes
1 answer
293 views

how to evaluate $\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$

Question: how to evaluate $$\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$$ MY try to evaluate the integral $$ \begin{aligned} & I=\int_0^{\infty} \frac{x \...
user avatar
6 votes
3 answers
211 views

Closed form for $\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$?

Is there a closed form for $I=\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$ ? Context Earlier I asked "Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\...
Dan's user avatar
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2 votes
0 answers
85 views

Can multiary compositions of elementary functions have an elementary inverse?

I'm looking for general methods for solving equations of elementary functions of one variable in closed form. Definition: The elementary functions are generated by applying finite numbers of $\text{...
IV_'s user avatar
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1 vote
0 answers
46 views

Polylogarithmically solving $\int\frac{\log(a_1x+b_1)\cdots\log(a_nx+b_n)}{px+q}\,dx$

I am now trying a direct approach to solving my question about $$\int_0^\infty\frac{\arctan a_1x\arctan a_2x\dots\arctan a_nx}{1+x^2}\,dx$$ where the $a_i$ are all positive. Note that the $\arctan$s ...
Parcly Taxel's user avatar
3 votes
0 answers
51 views

Find $\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$ algorithmically

It sometimes happens that $$\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$$ is algebraic for positive integers $m,n,a_k,b_k$. For example, $$\frac{\Gamma\left(\frac{1}{24}\right)\Gamma\left(\...
Nomas2's user avatar
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2 votes
1 answer
76 views

How to evaluate $\int_1^{\infty}\frac{t^2\ln^2 t\ln(t^2-1)}{1+t^6}{\rm d}t $

I was evaluating Evaluate $\displaystyle\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 x){\rm d}x .$ On the path of integrating the main function, I am stuck at this integral. I don't know how ...
Martin.s's user avatar
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4 votes
3 answers
212 views

Show that $\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$

Problem: Show that $$\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$$ Some thinking before trying At least we ...
Martin.s's user avatar
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0 votes
0 answers
22 views

Is there a closed form for $\int _0 ^1 t^{a-1}(1+t)^{b-1}dt$? [duplicate]

$$B(a,b):=\int _0 ^1 t^{a-1}(1-t)^{b-1}dt$$ What happens If we change the negative sign to positive ? $$F(a,b):=\int _0 ^1 t^{a-1}(1+t)^{b-1}dt$$ This question came to me while solving this limit $$...
pie's user avatar
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7 votes
1 answer
256 views

Evaluate $\int_0^\infty\frac{dx}{1+x^2}\prod_i\arctan a_ix$ (product of arctangents and Lorentzian)

Define $$I(a_1,\dots,a_n)=\int_0^\infty\frac{dx}{1+x^2}\prod_{i=1}^n\arctan a_ix$$ with $a_i>0$. By this answer $\newcommand{Li}{\operatorname{Li}_2}$ $$I(a,b)= \frac\pi4\left(\frac{\pi^2}6 -\Li\...
Parcly Taxel's user avatar
0 votes
3 answers
73 views

How do I calculate the closed form of $\sum_{k=2}^\infty kx^{k-2}$

This is an exercise from Wade, the answer is given as; $$\sum_{k=2}^\infty kx^{k-2}=\frac{2-x}{(1-x)^2},$$ but there is no help as to how to arrive at that answer. I have completed the first question ...
MW1's user avatar
  • 3
3 votes
1 answer
123 views

Power series for $\sum_{n=0}^\infty(-1)^n/n!^s$ (around $s=0$)

I'm looking for ways to compute the coefficients of the power series $$ \sum_{n=0}^\infty\frac{(-1)^n}{n!^s}=\sum_{k=0}^{\infty}c_k s^k $$ (a prior version of the question asked whether such an ...
metamorphy's user avatar
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0 votes
0 answers
20 views

Solve Minimax Rules in Finite Case

Let $\Theta = \{\theta_1, \cdots, \theta_n\}$ be the space of parameters and $D = \{d_1, \cdots, d_m\}$ be the space of decisions (that is, they are arbitrary finite sets with at least two elements). ...
温泽海's user avatar
  • 2,400
3 votes
2 answers
86 views

Integral in terms of Hypergeometric function

Consider the integral $$ I = \int_0^1 \int_0^1 (1+c^2v^2)^{-s}u^{1-2s}(1-uv)^{s-1}dudv$$ where $c>0$ is some constant and $0<s<1$. Clearly the integral is absolutely integrable (Two ...
Sam's user avatar
  • 3,205
0 votes
0 answers
73 views

Is it possible to find a closed form for $i!$? [duplicate]

I am curious is there a closed form for $i!$? I tried to search for any closed form for this but I didn't find any. $$z! := \lim_{n \to \infty } n^z \prod_{k=1}^n \frac {k}{z+k}$$ $$i! =\lim_{n \to \...
Mathematics enjoyer's user avatar
12 votes
1 answer
616 views

Prove $\int_0^\pi\arcsin(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}})dx=\frac{\pi^2}{5}$.

There is numerical evidence that $$\int_0^\pi\arcsin\left(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}}\right)dx=\frac{\pi^2}{5}.$$ How can this be proved? Context In another question, three random ...
Dan's user avatar
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6 votes
0 answers
93 views

Is there a closed-form expression for this iterated mean?

Here is a simple Python implementation of the arithmetic, geometric, and harmonic means of a (non-empty) list of numbers: ...
Dan's user avatar
  • 15.5k
5 votes
2 answers
102 views

Simplify in closed-form $\sum_n P_n(0)^2 r^n P_n(\cos \theta)$

Simplify in a closed form the sum $$S(r,\theta)=\sum_{n=0}^{\infty} P_n(0)^2 r^n P_n(\cos \theta)$$ where $P_n(x)$ is the Legendre polynomial and $0<r<1$. Note that $P_n(0)= 0$ for odd $n$ and $...
bkocsis's user avatar
  • 1,258
6 votes
2 answers
242 views

Find the closed form of $_3F_2(\frac{1}{4},\frac{3}{4},\frac{5}{4};\frac{3}{2},\frac{7}{4};1)$

Context Some investigation suggests that the following identity is true: \begin{align} _3F_2(\frac{1}{4},\frac{3}{4},\frac{5}{4};\frac{3}{2},\frac{7}{4};1)=\frac{3\sqrt{2}\sqrt{\pi}\left(2\log({1+\...
User's user avatar
  • 359
4 votes
2 answers
232 views

Closed form for this generalisation of the gamma function. $f(x+1)=f(x)g(x+1) $

Just for curiosity I want to generalise the Pi function i.e $f(x+1) = f(x)g(x+1)$ for some differentiable function, I know this function probably has no closed form for general functions $g$ as I ...
pie's user avatar
  • 5,405
20 votes
1 answer
987 views

Prove $\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12$.

There is numerical evidence that $$I=\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12.$$ How can this be proved? ...
Dan's user avatar
  • 23.8k
6 votes
2 answers
137 views

How to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$?

Just for curiosity I want to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$. Since $x=P_1,\ xP_n= P_{n+1} +nP_n$, this proves that it is possible for any $x^n$ to be represented as a sum ...
pie's user avatar
  • 5,405
12 votes
2 answers
488 views

How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$?

How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine. Yes, I am aware there is no reason to believe a random power ...
Alma Arjuna's user avatar
  • 3,821
2 votes
1 answer
183 views

Property of $a_{n+1} = a_n - \frac{1}{a_n}$

For a $a_n$ defined recursively by $a_{n+1} = a_n - \frac{1}{a_n}$,$a_0 = k >0$. Prove that if the first $n$ such that $a_n \leq 0$, then $n \in O(k^2)$. I ran a computer simulation, and it seems ...
tovdan's user avatar
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