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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-3
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1answer
33 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
67 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
53 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
62 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 ...
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
254 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
190 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
183 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
votes
1answer
43 views
2
votes
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
436 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?
13
votes
1answer
169 views

Closed form for $_2F_1\left(\frac12,\frac23;\,\frac32;\,\frac{8\,\sqrt{11}\,i-5}{27}\right)$

I'm trying to find a closed form (in terms of simpler functions) for the following hypergeometric function with a complex argument: ...