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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40 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
188 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
34 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
168 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
78 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" ...
3
votes
0answers
87 views
+50

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
54 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
81 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
83 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
27 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
115 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
55 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
79 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
26 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
64 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
143 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 ...
2
votes
1answer
75 views

How to solve the following equation? $\left(\sqrt{u^2-1}+u\right)^{1/u}=\pi ^{1/\pi }$

I have no clue: $$\left(\sqrt{u^2-1}+u\right)^{1/u}=\pi ^{1/\pi }$$
1
vote
0answers
28 views

Closed form for $\sum_{k\in\mathbb{N}}\frac{k}{a\uparrow^kb}$

Let $a,b\in\Bbb{N}$. Is there a closed form for $\displaystyle\sum_{k\in\mathbb{N}}\frac{k}{a\uparrow^kb}$ ? (I use Knuth's up arrow notation) If so, how can we obtain it ? If there isn't a closed ...
1
vote
2answers
74 views

Solving 2nd order linear recurrence with non-constant coefficients

I am trying to find a general solution to the following definite integral: $$F_{n}{\left(a,b;z\right)}:=\int_{a}^{z}\frac{x^{n}}{\sqrt{\left(x-a\right)\left(b-x\right)}}\,\mathrm{d}x,\tag{1}$$ ...
1
vote
2answers
50 views

Equations involving factorial/Gamma function

Are there any known methods to formally solve equations like: 1)$x^3!+(2x^2)!-x!+3=0$ 2)$x!=e^x$ ($0$ is trivial but there must be another one) 3)$(2x!)^2+x!-1=0$ 4)$x!!+x!=7$ I don't need ...