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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8
votes
1answer
271 views

Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$

Prove: $$ {\Large\int_{0}^{\ln^{2}\left(2\right) \over4}}\, \frac{\arccos\left(\vphantom{\huge A} {\exp\left(\vphantom{\large A}\sqrt{x\,}\right) \over \sqrt{\vphantom{\large A}2\,}}\right)} ...
1
vote
3answers
195 views

Is there a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$?

I am interested in finding a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$. Does anyone know if there is some Binomial identity that might be helpful here? Thank you.
11
votes
2answers
267 views

Conjecture $\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$

$$\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$$ Is it possible to prove this?
22
votes
2answers
671 views

Are there other cases similar to Herglotz's integral $\int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt$?

This post of Boris Bukh mentions amazing Gustav Herglotz's integral $$\int_0^1\frac{\ln\left(1+t^{\,4\,+\,\sqrt{\vphantom{\large A}\,15\,}\,}\right)}{1+t}\ \mathrm ...
16
votes
0answers
237 views

Simplify $\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};1,\frac{3}{2};\frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\big|\frac{1}{\sqrt{3}}\right)}$

Is it possible to simplify the ratio $$\mathcal{E}=\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};\ 1,\frac{3}{2};\ \frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\Big|\frac{1}{\sqrt{3}}\right)},$$ ...
47
votes
2answers
1k views

Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx$

Consider the following integral: $$\mathcal{I}=\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx,$$ where ...
1
vote
1answer
64 views

Is it possible to get a 'closed form' for $\sum_{k=0}^{n} a_k b_{n-k}$?

This came up when trying to divide series, or rather, express $\frac1{f(x)}$ as a series, knowing that $f(x)$ has a zero of order one at $x=0$, and knowing the Taylor series for $f(x)$ (that is ...
7
votes
1answer
197 views

Integral $\int_0^\infty\frac{dx}{\frac{x^4-1}{x\cos(\pi\ln x)+1}+2x^2+2}$

I need your help with this integral: $$\int_0^\infty\frac{dx}{\frac{x^4-1}{x\cos(\pi\ln x)+1}+2\,x^2+2}.$$ I wasn't able to evaluate it in a closed form, although an approximate numerical evaluation ...
2
votes
2answers
104 views

Help calculating $\sum^{R}_{k=1} \bigl\lfloor{\sqrt { R^2-k^2}}\bigr\rfloor$

I'm trying to calculate, or at least approximate, $$\sum^{R}_{k=1} \left\lfloor{\sqrt { R^2-k^2}}\right\rfloor,$$ where $R$ is a natural number. I have tried factoring this as $$\sum_{k=1}^R ...
5
votes
1answer
150 views

A conjecture $\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx\stackrel?=\frac\pi2\ln2$

I need to find a closed form for this integral: $$\mathcal{I}=\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx.$$ A numerical integration results in an approximation ...
16
votes
2answers
444 views

A conjectured closed form of $\int_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$

Consider the following integral: $$\mathcal{I}=\int_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx.$$ I tried to evaluate $\mathcal{I}$ in a closed form (both manually and using ...
23
votes
2answers
348 views

A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$

Is it possible to evaluate the following integral in a closed form? $$\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx,$$ where $\phi$ is the golden ratio: ...
6
votes
2answers
174 views

Is there a closed form expression for $1 + x + x^{4} + x^{9}+x^{16}+x^{25} +..+x^{k^2}$?

We all know what the sum of a geometric series is $1 + x + x^2 + x^3 + ... + x^k = \frac{x^{k+1} - 1}{x - 1}$ . I was wondering if similar formulas exist in case the exponents form some other ...
2
votes
2answers
92 views

Find a good candidate for a closed-form solution of this recurrence relation: $P(n-1)+n^2$.

I want to find a candidate for this recurrence relation: $$ P(n) = \left\{\begin{aligned} &1 &&: n = 0\\ &P(n-1)+n^2 &&: n>0 \end{aligned} \right.$$ Starting from 0 the ...
2
votes
2answers
133 views

Closed form for $\int \frac{1}{x^7 -1} dx$?

I want to calculate: $$\int \frac{1}{x^7 -1} dx$$ Since $\displaystyle \frac{1}{x^7 -1} = - \sum_{i=0}^\infty x^{7i} $, we have $\displaystyle(-x)\sum_{i=0}^\infty \frac{x^{7i}}{7i +1} $. Is ...
22
votes
2answers
629 views

A conjectural closed form for $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$

Let $$S=\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!},\tag1$$ its numeric value is approximately $S \approx 0.517977853388534047...$${}^{[more\ digits]}$ $S$ can be represented in terms of the ...
2
votes
0answers
85 views

An example of Risch algorithm for integrating $y$, $F(x,y)=0$.

I would like to compute the integral $$ \int y\,dx, \qquad y=\sqrt{x+\sqrt{x+1}},\\ F(x,y)=y^4-2xy^2+x^2-x-1=0, $$ in closed form, where $F(x,y)$ is a polynomial in $\mathbb{C}[x,y]$. I am trying to ...
8
votes
2answers
184 views

Conjectural closed form for $\int_0^\infty\sqrt[3]z\ \operatorname{Ei}^2(-z)\,dz$

While trying to answer the question "A closed form for $\displaystyle\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx$", I came up with a conjecture: $$\int_0^\infty\sqrt[3]z\ ...
16
votes
2answers
219 views

A closed form for $\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx$

Let $\operatorname{li}x$ denote the logarithmic integral $^{[1]}$$^{[2]}$$^{[3]}$: $$\operatorname{li}x=\int_0^x\frac{dt}{\ln t}$$ and $$I=\int_0^1\frac{\ln(-\ln x)\ ...
1
vote
1answer
32 views

Minimization of $\text{tr} (W^TMW)-\text{tr}(NW)$ subject to $W^TW=I$

Is there a closed-form solution for finding W that minimizes the objective function: $\text{tr} (W^TMW)-\text{tr}(NW)$ subject to $W^TW=I$ where $M$ and $N$ are fixed matrices. I find it difficult to ...
1
vote
4answers
61 views

closed form $f_n=\sqrt{2f_{n-1}}$ ? [duplicate]

I am trying to write up a proof for the convergence of this recursive function. I was wondering if there exists a closed form. Given first term in sequence is $\sqrt{2}$ and second is ...
0
votes
3answers
88 views

How to solve a recursive equation

I have been given a task to solve the following recursive equation \begin{align*} a_1&=-2\\ a_2&= 12\\ a_n&= -4a_n{}_-{}_1-4a_n{}_-{}_2, \quad n \geq 3. \end{align*} Should I start by ...
20
votes
1answer
931 views

Integrating $\int_0^ex^{1/x}\;dx$

Compute $$\int_0^ex^{1/x}\;\mathrm dx.$$ There is an analytical anti-derivative found in this answer. How does one compute this? Using the anti-derivative approach we have $$\int x^{1/x}\;\mathrm d ...
19
votes
1answer
566 views

A definite integral $\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$

I need to find a value of this definite integral: $$\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx.$$ Its numeric value is approximately $0.7875720991394284$, and lookups ...
20
votes
2answers
483 views

Integral $\int_0^\infty\left(x+5\,x^5\right)\operatorname{erfc}\left(x+x^5\right)\,dx$

Is it possible to find a closed form (possibly using known special functions) for this integral? $$\int_0^\infty\left(5\,x^5+x\right)\operatorname{erfc}\left(x^5+x\right)\,dx$$ where ...
0
votes
1answer
77 views

Closed Forms of Certain Zeta constants?

The Riemann Zeta function $\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ converges for $Re(s)>1$. I am interested in some particular "irrational " Values of it. Like $\zeta(\pi)=1.176241738...$ ...
32
votes
4answers
1k views

An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}dx$

I need your help with this integral: $$\mathcal{K}(p)=\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}dx,$$ where $\operatorname{Ai}$, $\operatorname{Bi}$ are Airy functions: ...
6
votes
0answers
119 views

Need help with $\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$

Could you please help me to solve this integration problem? $$\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$$ Its approximate numeric value is $0.419197813818367...$, but I could not find an exact ...
23
votes
1answer
438 views

Integral $\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$

I encountered this scary integral $$\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$$ where $_3F_2$ is a generalized hypergeometric function ...
21
votes
6answers
792 views

Evaluating $‎\sum_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$

‎If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then‎, $$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)‎.$$ ‎Since ...
17
votes
1answer
237 views

A closed form for $\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]{2}\right)}{x}dx$

I am stuck with this integral: $$\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]{2}\right)}{x}dx,$$ where $J_0$ is the Bessel function of the first kind. Is it possible to express this ...
1
vote
1answer
62 views

Number of points in a Bresenham's circle

The Midpoint circle algorithm generates a set of quantized coordinates for a circle of a given radius. The number of points generated for is of course a multiple of $4$ due to symmetry, but I didn't ...
2
votes
4answers
427 views

How to find a closed form solution to a recurrence of the following form?

I need to find the closed form solution to the following recurrence -: $ T(n) = 8*T(n/2) + 0.25*n^2$ with $T(1) = 1$ and $n=2^j$ and this is what I have tried so far but just can't seem to get a ...
4
votes
0answers
96 views

closed form for $\int_{0}^{\infty}\text{Ci}(x)^ndx$ [closed]

$$\int_{0}^{\infty}\text{Ci}(x)^ndx$$ where $$\text{Ci}(x)=-\int_{x}^{\infty}\frac{\cos t}{t}dt$$ is cosine integral
17
votes
3answers
213 views

A closed form for $\int_0^\infty e^{-a\,x} \operatorname{erfi}(\sqrt{x})^3\ dx$

Let $\operatorname{erfi}(x)$ be the imaginary error function $$\operatorname{erfi}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{z^2}dz.$$ Consider the following parameterized integral $$I(a)=\int_0^\infty ...
16
votes
2answers
289 views

A closed form for $\int_0^\infty\frac{\sin(x)\ \operatorname{erfi}\left(\sqrt{x}\right)\ e^{-x\sqrt{2}}}{x}dx$

Let $\operatorname{erfi}(x)$ be the imaginary error function $$\operatorname{erfi}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{z^2}dz.$$ Consider the integral $$I=\int_0^\infty\frac{\sin(x)\ ...
0
votes
0answers
27 views

Writing an integer as a sum of four integers, subject to restrictions [duplicate]

Let $a_n$ be the possible number of ways to write $n=i_1+i_2+i_3+i_4$ where $i_1,...,i_4$ are all integers. Here's their restrictions: $$0<i_1<3\\i_2>0\\i_3=2\text{ or }4\\i_4>1$$ Write ...
0
votes
1answer
122 views

How to find a compact expression for $\sum\limits_{k=1}^n \frac k{(K+1)!}$?

$$\sum_{k=1}^n \frac k{(K+1)!}$$ How to find a compact expression? (Original scan here)
1
vote
1answer
100 views

Recursive and closed form solution for choosing $n$ pairs/triplets.. of $kn$ elements.

I stumbled apon an interesting question: How many ways are there to arrenge $kn$ elements into $n$ sets, $k$ elements each? There should be a recursive and closed form solution for $g_k(n)$. For ...
13
votes
3answers
263 views

Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$

Let $$f(a)=\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx,$$ where $\operatorname{sech}(z)=\frac2{e^z+e^{-z}}$ is the hyperbolic secant. Here are values of $f(a)$ at some ...
5
votes
1answer
237 views

Closed form of $\sum_{n=1}^\infty \frac{2^{3n}}{2^{4n}+2^{2n}} $?

$$\sum_{n=1}^\infty \frac{2^{3n}}{2^{4n}+2^{2n}} $$ I know it converges thanks to WolframAlpha, but want to know exactly what it converges to. I can calculate the decimal, but don't know how to begin ...
0
votes
1answer
102 views

Getting the closed form solution of a third order recurrence relation with constant coefficients

This is part of the proof of finding the closed from solution of third order recurrence relation I know that the closed form will look like the following And this is the part of the proof I can ...
0
votes
2answers
414 views

Finding the closed form solution of a third order recurrence relation with constant coefficients [duplicate]

How would you solve for the closed form solution of a(n) given the general form of the third order linear homogenous recurrence relation with real constant coefficients. ...
4
votes
1answer
111 views

Is there a closed form for $\sum_{i=0}^{\text{min}(k,t)} {{\binom k i} \over {\binom t i}}\cdot x^i$?

Is there a closed form for the following summation? ($k,t \in N, x\in \mathbb R^+$) $$f(x,k,t)=\sum_{i=0}^{\text{min}(k,t)} {{\binom k i} \over {\binom t i}}\cdot x^i$$
1
vote
0answers
60 views

Not closed under equality?

One of the Peano axioms state that "For any $a \in \Bbb N : a = b, b \in \Bbb N$." An example where transition is not closed under equality is the relation to friends. C may not be B's friend, but A ...
5
votes
3answers
298 views

Find inverse for the closed-form expression of linear recurrence relation

I am trying to find an inverse of the following formula: $$ a_n=\frac{2+\sqrt{6}}{4}(1+\sqrt{6})^n+\frac{2-\sqrt{6}}{4}(1-\sqrt{6})^n $$ This formula is a closed-form expression of a linear ...
11
votes
1answer
222 views

Determining the number of valid TicTacToe board states in terms of board dimension

I am attempting to find a closed form equation in terms of $n$, for the number of valid Tic-Tac-Toe board states (ignoring symmetry), where the board has dimension $n \times n ,\; 0 \lt n,\;n \in \Bbb ...
1
vote
1answer
109 views

Is there an algorithm to determine if a closed form solution exists?

My sister asked me something like five years ago to prove that a particular physical problem has a closed form solution. Are there some theorems to prove the existence of closed form solutions? The ...
4
votes
1answer
143 views

The homology of the torus.

I am reading "Riemann surface" by Donaldson. On page 68, the calculation of the first homology of the torus $T$ is given but there are several steps that I don't understand. Here is the calculation. ...
78
votes
2answers
3k views

How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$

How can one prove that $$\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}$$ $$=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$$