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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2
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0answers
22 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}^{+}\land0<a<1$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
2
votes
1answer
40 views

Explicit solution of parametric solutions of an ODE

I need to find the explicit solution of the following ODE: $y'+\sin y'=x$, $y=y(x)$. I have found these two parametric solutions: $x=t+\sin t$ and $y=\frac{t^2}{2}+t\sin t+\cos t+c$, $c\in\Bbb R$. ...
-2
votes
2answers
41 views

How to solve this equation to find a closed-from for x? [on hold]

I want to find the value of $x$, i.e., $x=$ $$Ax+10^{-Bx}=C$$ Any suggestions?
-1
votes
0answers
13 views

Can some one help me to find a closed-from expression of X.

I want to get $X=\cdots$ $$A=10-10\cdot\log_{10}\left(10^{-B/10}\cdot\left(7.5+X-\frac{1}{1.8\cdot \ln(10)}\right)+\frac{1}{1.8\cdot \ln(10)}\cdot10^{-1.8 X}+10^{-C/10}\right)$$ That is if $A$, ...
3
votes
0answers
43 views

Closed form of an infinite series of integrals $\int_{0}^{\eta} \cos nt \cos t \sqrt{\cos^2 t - \cos^2 \eta}$

Let $$ I(n,\eta) = \int_{0}^{\eta} \cos nt \, \cos t \, \sqrt{\cos^2 t - \cos^2 \eta}\; dt $$ where it is known that $0 < \eta \leq \frac \pi 2$. Is it possible to evaluate $S$, the infinite ...
13
votes
3answers
201 views

Integral involving Clausen function ${\large\int}_0^{2\pi}\operatorname{Cl}_2(x)^2\,x^p\,dx$

Consider the Clausen function $\operatorname{Cl}_2(x)$ that can be defined for $0<x<2\pi$ in several equivalent ways: ...
2
votes
3answers
47 views

Determine a closed form for this sequence

Every year, 38 % of the amount of fish in a pond die. The 1st of May 2011 there were 5200 fish in the pond. Every year after May 1st 2011, 1900 new fish are added to the pond. Let $a_n$ be the amount ...
7
votes
3answers
175 views

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

$$\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$$ I have difficulty to evaluating above integrals. First i try the subsititue $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes ...
0
votes
1answer
39 views

Does the closed form of $f(t) = \int \frac{e^{2 \pi i \alpha t}}{e^{2 \pi i \beta t} - 1} dt$ exist?

I have been working on finding close forms of various Fourier series. The general approach is: From the series find the (not necessarily homogeneous) ordinary differential equation for which the ...
3
votes
2answers
81 views

Computing $\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$.

I would like to compute the following integral : $$\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$$ using Residue theorem. I took the contour corresponding to half of the "donuts" ...
5
votes
0answers
130 views

Can these integrals be represented in closed form?

This paper in the formula F.3.6 (page 271) gives the following formula for the derivative of Hurwitz Zeta function: $$\frac ...
2
votes
0answers
55 views

Closed form for sequence: $\sum_{j=1}^k 2^j j^{1/2}$

Any idea how to find the closed form for either of following sequences: $$ A(k) = \sum_{j=1}^k 2^j j^{1/2} $$ or $$ B(k) = \sum_{j=1}^k 2^a, \quad \quad a:= {j^{1/2}} $$ Note: closed form for one of ...
-1
votes
0answers
19 views

Closed form for $\sum _{k=r}^s \binom{n}{k}$

The cardinality of the Powerset is $2^n$. Looking for $\sum _{k=r}^s \binom{n}{k}$, Mathematica gives $$\sum _{k=r}^s \binom{n}{k}=\binom{n}{r} \, _2F_1(1,r-n;r+1;-1)-\binom{n}{s+1} \, ...
3
votes
1answer
46 views

Closed forms for definite integrals involving error functions

I have been working for a while with these kinds of integrals $$\int_0^\infty dx\,\text{erfc}\left(c +i x\right)\exp \left(-\frac{1}{2}d^2x^2+i cx\right)$$ $$\int_\Lambda^\infty ...
0
votes
2answers
30 views

Solving an inequality involving a floor

Increasing the integer $k$, I can make the floor of $L/k$ smaller than $r$: $$\left\lfloor \frac{L}{k} \right\rfloor \lt r$$ where $L, k, r$ are positive integers, $k\leq \lfloor \frac{L}{2} ...
13
votes
2answers
200 views

Closed form $\int_{-1}^{1} \frac{\ln (\sqrt{3} x +2)}{\sqrt{1-x^{2}} (\sqrt{3} x + 2)^{n}}\ dx$

Does the following integral $$\int_{-1}^{1} \frac{\ln (\sqrt{3} x +2)}{\sqrt{1-x^{2}} (\sqrt{3} x + 2)^{n}}\ dx, \; \; n \in \mathbb{N}$$ have a nice closed form? Basically I cannot tackle it in any ...
1
vote
3answers
82 views

Closed form of a sum of ratios of integers

I am computing in a program this sum (does it have a "name"): $$\sum_{\alpha=2}^{K} \frac{\alpha-1}{\alpha}$$ is there a way to avoid the sum, term by term, and use a more compact closed form ?
11
votes
3answers
136 views

Finding this summation: $\sum_{n=1}^{\infty}\frac{(2n+99)!(3n-2)!}{(2n)!(3n+99)!}$

What would be an easy method to find the approximate value of/close the form (the former will work too, if reasonably correct, a few decimal places, not more than that.): ...
5
votes
1answer
84 views

A difficult integral of improper integral

Could you help me calculate the integral? $$\int_0^{ + \infty } {{e^{ - x}}\left( {\frac{1}{{x\left( {{e^{ - x}} - 1} \right)}} + \frac{1}{{{x^2}}} + \frac{1}{{2x}}} \right)dx} .$$
1
vote
2answers
95 views

Closed expression of the following integral?

I believe that the following integral has a closed expression, but I haven't been able to check it $$I(k)=\int_{-\infty}^{\infty}dt\,\text{erf}\left(\frac{t}{b}-i \frac{1}{2}b(k+a)\right) ...
0
votes
0answers
20 views

Closed form for sum resembling generating function

Is there a general closed-form solution for $$\sum_{k=0}^\infty \frac{f(k)}{z-k}$$ as a generating function of $f$? It vaguely reminds of a couple of other kinds of generating functions, but not in ...
2
votes
2answers
55 views

How to obtain a closed form for summation over polynomial ($\sum_{x=1}^n x^m$)? [duplicate]

What is the method for obtaining the polynomial equal to \begin{equation*} \sum^{n}_{x=1}x^m \end{equation*} for unknown $n$, and systematically for various values of $m$? I know it should be a ...
5
votes
0answers
34 views

How do I find the finite limits of this infinite product?

What is... $$\lim_{\omega \to \infty} \left( {1 \over {a^{\omega}}} \cdot \prod_{N=1}^{\omega} (1+e^{b \cdot c^{-N}}) \right)$$ I'd like closed form solutions, and in this case that means any ...
2
votes
1answer
50 views

Closed-form expression for $\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$?

As per the title, I am looking for a closed-form expression for the integral $$\frac{1}{B(\alpha,\beta)}\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$$ where $a,\alpha,\beta>0$ and ...
0
votes
1answer
38 views

Probability that a random graph is connected

Let $V=\{v_1,\dots,v_n\}$ a set of $n$ vertices. Define $\mathcal{G}$ to be the set of all graphs on $V$. $|\mathcal{G}|=2^{\binom{n}{2}}$. What is the probability that a random graph from ...
1
vote
2answers
54 views

Calculate the sum of $\sum_{n=1}^{+\infty} \frac{1+2^n}{3^n}$

$$\sum_{n=1}^{+\infty} \frac{1+2^n}{3^n} = \sum_{n=1}^{+\infty} \frac{1}{3^n} + \sum_{n=1}^{+\infty} \frac{2^n}{3^n}$$ Each term is geometric series with $-1<r<1$ so they are all covergent. As ...
0
votes
3answers
28 views

Linear recursive sequence in closed-form function

I've been trying to find an answer for a question for some time, and I've done some Google searching but can't seem to figure out exactly how to solve it. It is a linear recursive sequence, and it ...
1
vote
4answers
124 views

Closed Form Expression of sum with binomial coefficient

I have the following equation which is making me problems. $$A_{n} = \sum_{k=0}^{n} \binom{n-k}{k}(-1)^{k}$$ where $n\in\mathbb{N}$. The task is to find a closed form expression for $A_{n}$. I have ...
3
votes
0answers
34 views

Does a better form exists for the coefficients of this product of power series?

Let $$f(a,b,t) = \sqrt{1-at}\sqrt{1-bt}$$ We take the series for $\sqrt{1-at}$ and $\sqrt{1-bt}$ around $t = 0$ and multiply them together to find $$f(a,b,t) = \sum_{n=0}^\infty ...
1
vote
2answers
117 views

How to solve equation $ x=W(a+bx^{n})+1 $?

How i can resolve the equation $x=W(a+b x^n)+1$, where $W$ is the Lambert $W$ function? thanks
1
vote
3answers
63 views

Evaluate the definite integral $ \int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx$

Evaluate the integral: $\displaystyle \int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx$ (using substitution) Here's my attempt at solution: u = $\sin^5(x)$ $du = 5\sin^4(x) \cdot \cos(x) ...
1
vote
1answer
32 views

Closed Form Summation Example

$$ \sum_{i=1}^n (ai +b) $$ Let $n \geq 1$ be an integer, and let $a,b > 0$ be positive real numbers. Find a closed form for the following expression. In other words you are to eliminate the ...
1
vote
1answer
22 views

Summation to Closed Form conversion

I am struggling to understand basics as it related to forming a closed form expression from a summation. I understand the goal at hand, but do not understand the process for which to follow in order ...
2
votes
1answer
22 views

Sum identity involving sin

How one can prove that $$\sum_{k=1}^n(-1)^k\sin(2k\theta)=\cos(n\pi/2+\theta+n\theta)\sec\theta\sin(n\pi/2+n\theta)?$$ It looks difficult as there is sum on the other side and product of trigonometric ...
3
votes
2answers
62 views

Find the closed-form for $\sum_{i=0}^n(-1)^i(\frac{1}{2})^i$

I start with simplifying: $$\sum_{i=0}^n(-1)^i(\frac{1}{2})^i=\sum_{i=0}^n(-\frac{1}{2})^i$$ then: $$S = 1 + (-\frac{1}{2}) + (-\frac{1}{2})^2 + ... +(-\frac{1}{2})^n$$ $$(-\frac{1}{2})S = ...
1
vote
0answers
53 views

Is there a closed form expression for the following definite integrals?

I am looking for a closed form for these two integrals $$\int_{-\infty}^{-a}\text{d}x \frac{1}{|x|}e^{-\frac{1}{2}x^2\sigma^2}e^{i k |x|}+\int_a^{\infty}\text{d}x ...
2
votes
2answers
56 views

Evaluate $-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$ in $\gamma$.

Evaluate $\gamma$ expressed, involving Lambert function, by $$-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$$ where $\gamma<1$. I doubt that it is possible to find a value for ...
2
votes
1answer
80 views

Richard Pavlicek's combinatorial problem

In the game of bridge, a standard deck is dealt to four players, 13 cards each. That gives a total of $\binom{52}{13,13,13,13}$ distinct deals. How many distinct deals can be dealt if all spot cards ...
1
vote
1answer
27 views

Closed-form expression of $\frac{1}{N^{n}} \left[ N^{n+1} - \sum_{k=1}^{N} (k-1)^{n} \right] $

Is there a nice closed-form expression for $$\frac{1}{N^{n}} \left[ N^{n+1} - \sum_{k=1}^{N} (k-1)^{n} \right] $$ where $n, N, k \in \mathbb{N}$. I can obtain an approximation for this for large ...
5
votes
1answer
58 views

show that $\int_0^{\infty}\ln(2x)\frac{ax-u(1-e^{-2ax})}{\sinh^2(ax)}x^{2u-1}dx=\frac{1}{2u}(\frac{\pi}{a})^{2u}|B_{2u}|:u \in N$

$$\int_0^{\infty}\ln(2x)\frac{ax-u(1-e^{-2ax})}{\sinh^2(ax)}x^{2u-1}dx=\frac{1}{2u}(\frac{\pi}{a})^{2u}|B_{2u}|:u \in N$$ using real or complexe analysis where $B_{2u}$ is bernoulli number
3
votes
0answers
65 views

About the closed form for $\lim_{y\to +\infty}\left(-\frac{2}{\pi}\log(1+y)+\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx\right)$

Recently, when facing a baby Rudin's exercise, I proved that: $$ \int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx = \frac{2}{\pi}\log(1+y)+O(1) $$ holds by integration by parts. Now I wonder if ...
5
votes
3answers
241 views

How to solve $\int_0^{\frac{\pi}{2}}\frac{x^2\cdot\log\sin x}{\sin^2 x}dx$ using a very cute way? [closed]

Few days ago my friend gave me this integral and i cant get how to solve this. The integral is:$$\int_0^{\large \frac{\pi}{2}}\frac{x^2\cdot\log{{\sin{x}}}}{\sin^2{x}}dx$$
4
votes
1answer
146 views

Help finding a closed form

I have the following function: $$\frac{2e^x}{e^{2x}+1+2x}=\sum_{n=0}^\infty \varepsilon_n\frac{x^n}{n!}$$ I would like to find a closed form for the $\varepsilon_k$. One thing that I do know is that ...
0
votes
2answers
80 views

Generating function and its closed form

Consider the inequality $x_1 + x_2 + x_3 + x_4 ≤n$ where $x_1,x_2,x_3,x_4,n ≥ 0$ are all integers. Suppose also that $x_2 ≥ 2$, that $x_3$ is a multiple of 4, and $1 ≤ x_4 ≤ 3$. Let $c_n$ be the ...
0
votes
1answer
36 views

Does this sum of products of binomial coefficients have a simple closed form?

Let $c,m,k$ be positive integers. Is there a simple closed form for the following sum? $$ \sum_{i=1}^{c-1} (-1)^i {c \choose i} {im \choose k} $$ Mathematica finds nothing, and Maxima's implementation ...
1
vote
0answers
56 views

Elusive closed form for card permutation problem

Does a closed form formula f(n) exist for the two rightmost columns? The two question marks are meant to be 0. The diagram is a summary of the numerical results from original question: Permutations ...
6
votes
2answers
193 views

Closed form of the sum $\sum\limits_{n=0}^\infty \exp(-n^3)$

I am trying to calculate the sum of the series $$\sum_{n=0}^\infty \exp(-n^3)$$ Can it be expressed in terms of known mathematical functions?
0
votes
0answers
17 views

Approximate distribution of product of N normal i.i.d.?

Given $N>30$ i.i.d. $X\approx\mathcal{N}(\mu_X,\sigma_X^2)$, looking for: accurate closed form distribution approximation of $Y=\prod_{n=1}^{N}{X}$ asymptotic normal approximation of same ...
0
votes
1answer
56 views

Closed form for nth term - generating functions

I think I am mostly confused about what the question is asking. I read that "closed form" means that it should not be represented as as infinite sum, so I am not sure what they are asking for. Would ...
2
votes
2answers
93 views

Closed form for an infinite sum over Gamma functions?

I am having quite a bit of trouble trying to find a closed form (or a really fast way to compute) for the infinite sum $$\sum_{n=1}^{\infty} a^n \dfrac{\gamma(n+1,b)}{\Gamma(n+1)\Gamma(n)}$$ where ...