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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1answer
32 views

Closed form questions [on hold]

Please could you help me to find the generating functions of the following sequences in closed form: (a) 1, 0, 1, 0, 1, 0, … (b) 2, –4, 6, –8, 10, –12, …
3
votes
1answer
62 views

Does this integral have any closed form? $\displaystyle\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$

Does this integral have any closed form? $$\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$$ I think the substitution $x=(u-1)+2\pi$ will do it, no?
0
votes
2answers
66 views

What is the sum of this series given the closed form?

The closed form of a series I am trying to identify is: $$ a_n=\frac{250}{2n -1} $$ How could I get the sum of the series equation from this? I am used to geometric sequences and arithmetic sequences ...
1
vote
0answers
13 views

Almost complete multivariate recurrence solution…

$$ \gamma c_{jm_1m_2} = s^+_{jm_1}c_{j(m_1+1)m_2} - s^+_{jm_2}c_{jm_1(m_2+1)}\\ \gamma d_{jm_1m_2} = s^-_{jm_1}d_{j(m_1-1)m_2} - s^-_{jm_2}d_{jm_1(m_2-1)} $$ where $s_{jm_i}^{\pm} = ...
3
votes
2answers
85 views

How could I have found the closed form of $\sum_{k=1}^n \frac{k}{(k+1)!}$ in advance?

If you calculate the first three sums, a pattern becomes clear revealing the closed form which is easily proven by induction: $$\sum_{k=1}^n \frac{k}{(k+1)!} = \frac{(n+1)!-1}{(n+1)!}$$ I’ve been ...
0
votes
0answers
40 views

Is there a closed form of the following expression.

Does anyone know of a way to write a closed form of the following expression using only addition, subtraction, multiplication, and division? $$ \left[\sum\limits_{i=1}^n \ln(i)\right]^e $$
2
votes
2answers
57 views

Closed form for $\prod_{k=1}^n \binom{k^2+2k}{k^2}$

Does anybody know how I can find a closed form for the expression $$ \prod_{k=1}^n \binom{k^2+2k}{k^2}? $$ There are many ways to handle such things, but with sum instead of product. Here, I have no ...
2
votes
1answer
58 views

Integration of $\int_0^\pi\int_0^\pi[\sin x\sin y\ge a]\,dx\,dy$

Does the integral $$I(a)=\int_0^\pi\int_0^\pi[\sin x\sin y\ge a]\,dx\,dy$$ have any closed-form solution? This is the area under the contours of $\sin x\sin y$; $[\cdot]$ is the Iverson bracket, which ...
0
votes
0answers
52 views

Evaluating $\int_2^\infty \zeta(x) - 1 \,\, \mathrm{d}x$

While looking at a table of values for the zeta function, the fact that they approach $1$ made me wonder what the improper integral of the fractional part of the zeta function would be. I've found ...
8
votes
1answer
163 views

Closed form for $\int_0^{\pi/2}\frac{\sqrt{1+\sin\phi}}{\sqrt{\sin2\phi}\,\sqrt{\sin\phi+\cos\phi}}d\phi$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\frac{\sqrt{1+\sin\phi}}{\sqrt{\sin2\phi}\,\sqrt{\sin\phi+\cos\phi}}d\phi$$ Its approximate numeric value is ...
1
vote
4answers
61 views

Probability of choosing a subset of elements where each element has a different probability

I am trying to write a C++ program to do this but nobody on Stackoverflow can seem to help me so I thought I'd try to do it myself with some help from you guys. My post on Stackoverflow can be found ...
5
votes
1answer
88 views

Closed form of $\int_0^\infty \frac{\log(x)-\log(a)}{x-a}e^{-x} \mathrm{d}x$.

Can the integral $$ \int_0^\infty \frac{\log(x)-\log(a)}{x-a}e^{-x} \mathrm{d}x $$ be expressed in terms of some simple special function? I have searched through integral tables but couldn't find ...
8
votes
0answers
81 views

Need help with $\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$

Please help me to evaluate this integral: $$\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$$
1
vote
2answers
41 views

Closed form of $T(n)=T(\lceil n/2 \rceil)+T(\lfloor n/2 \rfloor)+2$

How in God's name could I find a closed form of $T(n)=T(\lceil n/2 \rceil)+T(\lfloor n/2 \rfloor)+2$? I'm looking at the first numbers in sequence and I just don't see any relation...
0
votes
1answer
178 views

How to calc $\min ||J\Delta\tau + D||_*$

How to calculate $$ \min_{\tau} ||J_1 \tau_1 + \cdots + J_p \tau_p + D ||_* $$ where $\tau_1, \cdots, \tau_p \in \mathbb{R}$ $J_1, \cdots, J_p, D \in \mathbb{R}^{m \times n}$ $||\cdot||_*$ is sum ...
5
votes
2answers
65 views

Closed form of generating function consisting of power of two binomials

Let $g(x)$ be infinite formal power series and $$g(x) = (1 + x)(1 + x^2)\cdots(1 + x^{2^k})\cdots$$ Show that $(1 - x) g(x) = 1$. My book gives following proof: Using a fact that $(1 - x^k)(1 + ...
0
votes
0answers
39 views

Is there a way to simplify Legendre-squared sum $\sum_{n} \frac{[P'_{n}(x)]^2}{n(n+1)}$

Is there a closed-form expression for $$F(x) = \sum_{n=2,{\rm even}}^{\infty} \frac{[P'_{n}(x)]^2}{n(n+1)}$$ where $P'_{n}(x)$ is the derivative of the $n^{\rm th}$ Legendre polynomial? Simple ...
0
votes
1answer
26 views

Close form of a power series starting at $n=2$

This is the power series I am looking at $\sum_{n=2}^{\infty}{n(n-1)z^n}$. I want to find the closed form of this power series. This is my approach, if I divide the power series by $z^2$, then I ...
4
votes
3answers
182 views

Find a closed form from the given power series

I have the power series $\sum_{n=0}^{\infty} {z^{2n}\over{n!}}$, how do I find the closed form for this power series. I am aware that $e^z=\sum_{n=0}^{\infty} {z^{n}\over{n!}}$, so I tried to ...
0
votes
0answers
47 views

Look for closed form solution to the simple ODE

I'm looking for a possible closed form solution to the following ODE: $\dfrac{dy}{dt} = \log\left(1+\dfrac{c}{y}\right) y$ Thanks a lot in advance for the help.
3
votes
3answers
137 views

Conjectured closed form of $G^{2~2}_{3~3}\left(1\middle|\begin{array}c1,1;b+1\\b,b;0\end{array}\right)$

In my answer to this question, I come across the following case of the Meijer G-function: $$F(b)=G^{2~2}_{3~3}\left(1\middle|\begin{array}c1,1;b+1\\b,b;0\end{array}\right), b>0$$ and based on my ...
-1
votes
0answers
38 views

Closed Form of an X^3 equation

I try to find a closed form solution of this equation by using integral method, but is there really an answer to this? if there is no answer then why? and why do an equation can have no closed form? ...
11
votes
1answer
146 views

Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$

Let $\operatorname{Ei}x$ denote the exponential integral: $$\operatorname{Ei}x=-\int_{-x}^\infty\frac{e^{-t}}tdt.\tag1$$ It's not difficult to find that ...
0
votes
2answers
34 views

help defining an indicator function?

Consider some set: $A = \{1,2,3,4,5\}$ And a specific number, like $3$ I'd like some function $$f(a)=\begin{cases} 1 &\quad a>3\\0&\quad \text{otherwise}\end{cases}$$ - i.e. $f(4)=1,\ ...
13
votes
2answers
250 views

Crazy $\int_0^\infty{_3F_2}\left(\begin{array}c\tfrac58,\tfrac58,\tfrac98\\\tfrac12,\tfrac{13}8\end{array}\middle|\ {-x}\right)^2\frac{dx}{\sqrt x}$

Is there any chance to find a closed form for this integral? $$I=\int_0^\infty{_3F_2}\left(\begin{array}c\tfrac58,\tfrac58,\tfrac98\\\tfrac12,\tfrac{13}8\end{array}\middle|\ ...
13
votes
1answer
203 views

Formula for $\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$

Is it possible to express the following integral in terms of known special functions? $$I(a,b)=\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$$ I have managed to solve the special ...
15
votes
2answers
189 views

Integral $\int_0^1\frac{\ln x}{x-1}\ln\left(1+\frac1{\ln^2x}\right)dx$

Is it possible to evaluate this integral in a closed form? $$ I \equiv \int_{0}^{1}{\ln\left(x\right) \over x - 1}\, \ln\left(1 + {1 \over \ln^{2}\left(x\right)}\right)\,{\rm d}x $$ Numerically, ...
8
votes
1answer
152 views

Closed Form for $\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$

Is there a closed form for the following integral? $$\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$$ It is approximately equal to $-0.48878092308456029189008$. Mathematica is ...
8
votes
1answer
182 views

Strange closed forms for hypergeometric functions

So in the process of trying to find a derivation for this answer, the following interesting equalities arose (one can check with Wolfram Alpha/Mathematica): $$\frac{8\sqrt{2}G^4}{5\pi^2} ...
1
vote
2answers
50 views

Prove equality $a^{\log_b c} = c^{\log_b a}$

I'm try to prove the equality: $$a^{\log_b c} = c^{\log_b a}$$ I'm having trouble finding information regarding this, also I need to figure out why $n^{\log_2 3}$ is better than $3^{\log_2 n}$ as a ...
1
vote
0answers
53 views

Integral $\int_0^{\infty } \frac{1}{(\alpha x^2 + 1) \left(- 2 \sqrt{\frac{ x^2}{x^2+1}}+2 x+\pi \right)} \, dx$

Does the following integral admit a closed-form expression? $$\int_0^{\infty } \frac{1}{(\alpha x^2 + 1) \left(- 2 \sqrt{\frac{ x^2}{x^2+1}}+2 x+\pi \right)} \, dx \;\; , \;\; 0 \leq \alpha \leq ...
25
votes
2answers
463 views

Integral $\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx$

I'm struggling with this integral $$I=\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx.\tag1$$ Mathematica could not evaluate it in a closed form. Its numeric value is ...
0
votes
1answer
56 views

How to solve this summation (Lerch Transcendent)?

How is it possible to deduce the closed form of the following? $$\sum_{i = 0}^{n - 1} \frac{2^i}{n - i} = ?$$
17
votes
4answers
214 views

Integral $\int_0^1\frac{\log(1-x)}{\sqrt{x-x^3}}dx$

I have a trouble with this integral $$I=\int_0^1\frac{\log(1-x)}{\sqrt{x-x^3}}dx.$$ Could you suggest how to evaluate it?
6
votes
2answers
236 views

Transition time in a Lotka-Volterra system

I am working with a set of real-valued ordinary differential equations based on the Lotka-Volterra competition equations: $$\begin{align} \dot{a_1} & = a_1 \left( 1 - a_1 - 2 a_2 \right) \\ ...
1
vote
0answers
56 views

Closed-form expression of a definite integral

Does this definite integral admit a closed-form in terms of elementary functions? $$\int_0^{\infty } \frac{x}{\left(x^4+1\right) \left(2 x^2-2 \arctan\left(x^2\right)+\pi \right)} \, dx.$$
1
vote
1answer
23 views

Analytic expression for zeroes of sum of two sinusoids

I'm after a closed-form expression for the zeroes of the following function $$ p(z) = d_1 d_2 + d_1\cos(k_1 z) + d_2\cos(k_2 z) $$ $d_1$, $d_2$, $k_1$ and $k_2$ are all real constants. I'm after the ...
0
votes
0answers
33 views

Closed-form expression for a hypergeometric series

What is the closed-form expression for $${}_2 F_1 \left(1+2\lceil n/2\rceil,-n;1/2;-z/4\right)$$ According to the book Concrete Mathematics (R.Graham, D.Knuth, O.Patashnik 2nd), the authors say the ...
0
votes
2answers
97 views

Does this series $2 + 4 + \cdots + \sqrt{\sqrt{n}} + \sqrt{n} + n$ have a general term?

Does this sum simplify to a general term in terms of $n$? If so, how would you arrive at that term? $2 + 4 + \cdots + \sqrt{\sqrt{n}} + \sqrt{n} + n$. Thanks.
-1
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1answer
43 views
2
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3answers
59 views

Is there a closed form?

Is there a closed form for $k$ in the expression $$am^k + bn^k = c$$ where $a, b, c, m, n$ are fixed real numbers? If there is no closed form, what other ways are there of finding $k$? Motivation: ...
0
votes
1answer
67 views

No closed form proof

Prove that there is no closed form of the inverse of the expression $y = x\cot \frac{\pi }{x}$ where $x \geq 3$.I am currently completely lost. Thanks.
0
votes
1answer
26 views

Give exact value of s as a function of n in closed form

here is what I did so far but I cannot go further
3
votes
1answer
70 views

Homework: calculation about differential form

Here is the question: Let $\omega = A dy\wedge dz + B dz \wedge dx + C dx \wedge dy$ in $\mathbf{R}^3$, and $d\omega = 0$. Denote \begin{eqnarray} \alpha = \int_0^1 tA(tx,ty,tz)dt\cdot(ydz-zdy)\\ ...
10
votes
1answer
147 views

How to prove $\int_1^\infty\frac{K(x)^2}x dx=\frac{i\,\pi^3}8$?

How can I prove the following identity? $$\int_1^\infty\frac{K(x)^2}x dx\stackrel{\color{#B0B0B0}?}=\frac{i\,\pi^3}8,\tag1$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind: ...
11
votes
1answer
244 views

How to prove $\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right)$?

How can I prove the following conjectured identity? $$\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi\stackrel?=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right),\tag1$$ where $K(x)$ is the ...
16
votes
4answers
435 views

Closed form for $\int_{-1}^1\frac{\ln\left(2+x\,\sqrt3\right)}{\sqrt{1-x^2}\,\left(2+x\,\sqrt3\right)^n}dx$

I'm trying to find a closed form for the following integral: $$\mathcal{J}(n)=\int_{-1}^1\frac{\ln\left(2+x\,\sqrt3\right)}{\sqrt{1-x^2}\,\left(2+x\,\sqrt3\right)^n}dx\tag1$$ I have conjectured values ...
11
votes
1answer
87 views

How to evaluate $\int_0^\infty\frac{\frac{\pi^2}{6}-\operatorname{Li}_2\left(e^{-x}\right)-\operatorname{Li}_2\left(e^{-\frac{1}{x}}\right)}{x}dx$

I need to evaluate the following integral with a high precision: $$I=\int_0^\infty\frac{\frac{\pi^2}{6}-\operatorname{Li}_2\left(e^{-x}\right)-\operatorname{Li}_2\left(e^{-\frac{1}{x}}\right)}{x}dx,$$ ...
2
votes
0answers
55 views

Find the sum of exponentails of squares $\sum_{r=1}^n e^{-\alpha r^2}$

I would like to find $$a_n =\sum_{r=1}^n e^{-\alpha r^2},\qquad \alpha\in\mathbb{R}$$ I tried to solve the equivalent recursion $$a_n=a_{n-1}+e^{-\alpha n^2}\quad(n>0),\qquad a_0=0.$$ with an ...
0
votes
2answers
44 views

Expressing $\mathrm{B}(\sinh(x), \cosh(x))$ in terms of elementary functions

Is it possible to express: $\mathrm{B}(\sinh(x), \cosh(x))$ (where $\mathrm{B}$ is the beta function) In closed form, in terms of elementary functions?