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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13
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2answers
371 views

Evaluating $\int_0^{2\pi}\frac{dt}{\sqrt[4]{P(\cos t,\sin t)}}$

$${\LARGE\int}_0^{2\pi}\frac{dt}{\sqrt[{\LARGE 4}]{A\Big(\sin^8t+\cos^8t\Big)+B\Big(\sin^6t\cos^2t+\sin^2t\cos^6t\Big)+C~\sin^4t\cos^4t}}~=~?$$ where $A=0.3$, $B=-3.3$, and $C=10$. Its numerical ...
0
votes
2answers
85 views

A logarithm integral

Calculate the integral \begin{align} \int_{0}^{1} \frac{ \ln(\sqrt{x} - \sqrt{1-x}) }{ \sqrt{x} } \ dx \end{align} and show the value is negative.
0
votes
2answers
52 views

$x^x-x+5=\frac{29}{4}$

A friend of mine is claiming to have a closed form solution to $x^x-x+5=\frac{29}{4}$, plotting it into wolfram alpha gives an approximation, and the equation doesn't seem very easy to solve. Can any ...
4
votes
3answers
191 views

Harmonic Numbers series I

Can it be shown that \begin{align} \sum_{n=1}^{\infty} \binom{2n}{n} \ \frac{H_{n+1}}{n+1} \ \left(\frac{3}{16}\right)^{n} = \frac{5}{3} + \frac{8}{3} \ \ln 2 - \frac{8}{3} \ \ln 3 \end{align} where ...
3
votes
0answers
44 views

Dilogarithm in closed form

Is there a closed form expression for \begin{align} e^{\Large\frac{i\pi}3} \text{Li}_{2}\left( \frac{e^{\Large\frac{i\pi}3} }{2}\right) + e^{-\Large\frac{i\pi}3} \text{Li}_{2}\left( ...
0
votes
1answer
73 views

Summing up numbers from the continued fraction of $e ^ \pi$ and $\pi ^e$

I don't remember it well ,but it was around 5-6 years ago , I was 8 and I had found this new interest - continued fractions .I used to play with their terms sum them up and thought of getting ...
0
votes
0answers
37 views

Are generating functions ever analytic for logarithmic series?

Given a series $s_n = \ln(n) f(n)$ where $f(\cdot)$ is an elementary analytic function which does not involve the logarithm. More precisely $f$ can have simple poles but no branch cuts or essential ...
1
vote
1answer
63 views

May I know that there is a special structure or solution on this linear fractional optimization?

I am not familar with the optimization problems, but I want to know a very simple formulation: \begin{array}{cc} {{\max_\mathbf{x}}} & \frac{\mathbf{a}^{T}\mathbf{x}}{\mathbf{b}^{T}\mathbf{x}}\\ ...
8
votes
3answers
128 views

Do these integrals have a closed form? $I_1 = \int_{-\infty }^{\infty } \frac{\sin (x)}{x \cosh (x)} \, dx$

The following integrals look like they might have a closed form, but Mathematica could not find one. Can they be calculated, perhaps by differentiating under the integral sign? $$I_1 = \int_{-\infty ...
3
votes
0answers
70 views

Log Log Integrals III

The integrals \begin{align} I_{7} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( \ln \left(\frac{1}{x}\right) \right) \ \frac{dx}{1-x} \end{align} and \begin{align} I_{8} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( ...
0
votes
0answers
51 views

Closed form definite integral. [closed]

I have a definite integral that I am trying to solve. Any hint or reference is urgently sought $$\int_0^\infty \nu y^r e^{\nu y}\{1+\psi(e^{\nu y}-1)\}^{-\tfrac1\psi -1}dy.$$ where $r$ is any ...
2
votes
1answer
54 views

What is this waveform?

Consider the following infinite series: $\text{f} \left( x \right) =\displaystyle \sum \limits_{n=1}^{\infty} \frac {\sin \left( n x\right)}{n^2}$ We know that $\text{f} \left( x \right)$ is ...
9
votes
3answers
182 views

Definite integral - closed form: $\int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x$

I'm struggling with this definite integral: $$ \int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x. $$ Any help will be greatly appreciated.
1
vote
2answers
95 views

Log Log Integrals

Evaluate the integrals \begin{align} I_{1} &= \int_{0}^{1} \ln^{2n}(x) \ \ln\left(\ln\left(\frac{1}{x}\right)\right) dx \end{align} and \begin{align} I_{2} = \int_{0}^{1} \ln^{2n}(x) \ ...
0
votes
1answer
65 views

Finding the area enclosed by Batman curve

I was reading this thread: Is this Batman equation for real? and I wondered if the area bounded by the curve had a nice closed form. The resulting integrals were slightly beyond me, so I thought I ...
0
votes
2answers
66 views

Closed form of a sequence containing the pattern $\{+,-,-,+,+,-,-,…\}$

I was in the middle of answering this question that asked how to get the Taylor series of $cos(x)$ centered at $a=\frac{\pi}{3}$. I was reaching the point where I was about to write $(-1)^n$ making ...
1
vote
1answer
46 views

Matrix exponential of a simple bidiagonal matrix

I am interested in finding an expression (closed form or recursive) for the matrix exponential of this banded matrix: $$ \begin{pmatrix} 0 & 1 & 0 & 0 & \cdots & 0 & 0 ...
0
votes
0answers
29 views

Simple representation of a Sum

I have a probably pretty simple question. if I have $\sum_{i=0}^{n} 4^i$ and I want a closed representation, Wolfram Alpha gives me: $4^{n+1}/3-1/3$ Why is that? How do I get to that? Thanks!
1
vote
1answer
62 views

Is there closed form for $(1-p)(1-p^2)(1-p^3)…$ or its Taylor expansion?

I was considering the following problem: Somebody uses a backup for something (e.g. backups a file) and the backup is equally reliable as original storage. The storage is not perfectly reliable and ...
0
votes
1answer
44 views

Quadratic differential equation - closed form solution?

Can a solution for x(t) be found from the following real valued differential equation $$a\frac{d}{dt}\!(x(t))^2 + x(t) +b\frac{d}{dt}\!(y(t)) = 0$$ in terms of only y(t), it's integrals or ...
0
votes
1answer
23 views

Going from recurrence relations to closed form

How do I go from the following recurrence relation a(n) = (n+1)a(n-1) where a(0) = 2 to a closed form? I know I need to use an iterative approach but I am not ...
1
vote
3answers
111 views

Guess the closed form on the following sequence?

any help would be appreciated, have no idea where to start $u_1 = 2/3$ and $u_{k+1}$ such that: $$u_k + \frac{1}{(k+2)(k+3)}$$ for all, k are natural numbers guess a general formula (i.e the ...
0
votes
1answer
53 views

Closed form for nth term of generating function

How would I find the closed form for the $n^{th}$ term of a sequence? Is there a general formula I can follow for these types of problems? Taking this sequence for example... $$\frac{x^5}{(1-x)^4}$$
13
votes
3answers
298 views

$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$

I'm looking for a way to prove $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$ Since ...
-3
votes
1answer
47 views

Closed form questions [closed]

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, …
4
votes
1answer
87 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
70 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 ...
3
votes
2answers
90 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 ...
2
votes
2answers
67 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
87 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
66 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 ...
22
votes
2answers
565 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
79 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
101 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 ...
11
votes
2answers
199 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
51 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
184 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
72 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
58 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
40 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
254 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 ...
4
votes
3answers
153 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 ...
12
votes
1answer
185 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
38 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
671 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
250 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 ...
18
votes
2answers
363 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
167 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
206 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
53 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 ...