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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-4
votes
2answers
28 views

How to solve this equation to find a closed-from for x?

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

Sum of Harmonic numbers $\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$

Finding the closed form of: $$\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$$ where, $\displaystyle H_n^{(2)} = \sum\limits_{k=1}^{n}\frac{1}{k^2}$ It appears when we try to determine the ...
13
votes
3answers
197 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: ...
3
votes
0answers
42 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 ...
7
votes
3answers
174 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 ...
2
votes
1answer
72 views

Is it always true that no closed forms exists for any divergent series?

Having seen many questions regarding finding closed form of integrals or infinite series, and some users providing either the final answer or detailed solution, and also reading how one finds a closed ...
2
votes
3answers
35 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 ...
6
votes
2answers
243 views

Finite Series $\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})}$

I want to show that $$\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})} = \frac{n^2-1}6$$ With induction I don't know how I could come back from $\frac{1}{1-\cos(\frac{2k\pi}{n+1})}$ to ...
3
votes
2answers
79 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" ...
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
0answers
106 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 ...
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) ...
4
votes
1answer
452 views

Can Euler's reflection formula be used to calculate $\Gamma (1/3)$?

Can Euler's reflection formula be used to calculate $\Gamma (1/3)$? (That is, to express in in some closed form.) When trying to apply it, I am always falling in a loop.
15
votes
2answers
337 views

Improper Integral $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$

What is the closed form of the following integral for every $n\in\mathbb{N}$? $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$$ By Mathematica we see that $$\int_0^\frac{1}{2}x\cot(\pi ...
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 ...
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 ...
-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} \, ...
29
votes
4answers
1k views

Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

I want to find a closed form for this integral: $$I=\int_0^1\frac{\ln^2x}{\sqrt{x^2-x+1}}dx\tag1$$ Mathematica and Maple cannot evaluate it directly, and I was not able to find it in tables. A numeric ...
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 ...
5
votes
1answer
170 views

Closed form of $\int_{0}^{\pi/2}x\cot\left(x\right)\cos\left(x\right)\log\left(\sin\left(x\right)\right)dx$

I would like to know if there exists a closed form for this integral $$\int_{0}^{\pi/2}x\cot\left(x\right)\cos\left(x\right)\log\left(\sin\left(x\right)\right)dx.$$ I tried the relation ...
17
votes
6answers
635 views

Improper Integral $\int_{0}^{1/2}\left(2x - 1\right)^6\log^2\left(2\sin\pi x\right)\,dx$

How can I find a closed form for the following integral $$\int_0^{1/2}\left(2x - 1\right)^6\log^2\left(2\sin\pi x\right) \,dx$$
6
votes
3answers
163 views

A closed-form of product the gamma functions containing $\pi$ and $\phi$

Playing with gamma functions by randomly inputting numbers to Wolfram Alpha, I got the following beautiful result \begin{equation} ...
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} ...
3
votes
1answer
154 views

Generalized Logarithmic Integral - reference request

This page at I&S forum defines the Generalized Logarithmic Integral as $$L\left[ \begin{matrix} a,b,c \\ d,e,f \end{matrix};z\right] =\int_0^z \frac{\log^a x \log^b(1-x)\log^c(1+x)}{x^d (1-x)^e ...
8
votes
1answer
124 views

Infinite Series $\sum_{n=-\infty}^{\infty}\frac{1}{(x+n\pi)^m}$

How can we find a closed form for the following infinite series for any $m\in\mathbb N$? $$\sum_{n=-\infty}^{\infty}\frac{1}{(x+n\pi)^m}$$
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.): ...
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 ...
9
votes
2answers
207 views

Computing an indefinite integral

Let $\displaystyle P_n (x) = 1 + \frac{x}{1!} + \frac{x^2 }{2!} + \cdots + \frac{x^n }{n!} \ $ and $$ I(x) = \int \frac{2n!\sin x + x^n }{e^x + \sin x + \cos x + P_n (x)}\, dx $$ (where $\ n \to ...
16
votes
1answer
350 views

Infinite Series $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor$

How to prove that $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor=\gamma$$ Can we find a known value for ...
5
votes
1answer
84 views

A difficult integral of improper integral

Could you help me calculate the integral? $$\int_0^{ + \infty } {{e^{ - x}}\left( {\frac{1}{{x\left( {{e^{ - x}} - 1} \right)}} + \frac{1}{{{x^2}}} + \frac{1}{{2x}}} \right)dx} .$$
12
votes
2answers
261 views

Infinite Series $\sum_{n=1}^\infty\frac{\zeta(3n)}{2^{3n}}$

I want to evaluate $$\sum_{n=1}^\infty\frac{\zeta(3n)}{2^{3n}}$$ let $f(z)=\sum_{n=1}^\infty\frac{1}{2^{3n}}z^{3n}$, then $$\sum_{n=1}^\infty ...
4
votes
1answer
145 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 ...
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
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 ...
3
votes
3answers
109 views

Is there an algebraic solution to $e^{-x/a}+e^{-x/b}=1$ ($a\neq b$, $a,b$ constants)?

Is there an algebraic solution for the to find the intersection of the following two functions for values of $x\geq 0$: $$f_1(x)=1-2e^{-x/a}=f_2(x)=-1+2e^{-x/b}$$ $a$ and $b$ are positive constants. ...
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 ...
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 ...
30
votes
6answers
1k views

Infinite Series $‎\sum_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$

‎If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then‎, $$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)‎.$$ ‎Since ...
18
votes
1answer
282 views

Is $K\left(\frac{\sqrt{2-\sqrt3}}2\right)\stackrel?=\frac{\Gamma\left(\frac16\right)\Gamma\left(\frac13\right)}{4\ \sqrt[4]3\ \sqrt\pi}$

Working on this conjecture, I found its corollary, which is also supported by numeric caclulations up to at least $10^5$ decimal digits: ...
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 ...
11
votes
1answer
118 views

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. ...
4
votes
2answers
486 views

No closed form for the partial sum of ${n\choose k}$ for $k \le K$?

In Concrete Mathematics, the authors state that there is no closed form for $$\sum_{k\le K}{n\choose k}.$$ This is stated shortly after the statement of (5.17) in section 5.1 (2nd edition of the ...
0
votes
0answers
40 views

Euler type superdivergent

Could you explain where this came from: $$\sum _{k=0}^{\infty } (k!)^2 (-y)^k=\frac{G_{1,3}^{3,1}\left(\frac{1}{y}\mid{{0}\atop{0,0,0}}\right)+2 \left(\log \left(\frac{1}{y}\right)+\log (y)\right) ...
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 ...
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
15
votes
1answer
293 views

Infinite Series $\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}$

How can I prove that $$\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$$ I think this post can help me, but I'm not sure.
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 ...