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Questions tagged [closed-form]

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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Closed form for $q_k(0,0)$ from recurrence

We have $$p_0(n,m)=\begin{cases} 0,&\text{$n=m=0$}\\ (n-1)!,&\text{$n>0, m=0$}\\ 0,&\text{$n\geqslant0, m>0$} \end{cases}$$ $$q_0(n,m)=0, n\geqslant0, m\geqslant0$$ and $$p_k(n,m)=\...
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1answer
62 views

Prove $\sum_{k=1}^{\infty}\frac{(-1)^k}{(2k+1)^2}(2H_{2k}+H_k)=\frac{\pi^3}{32}-2G\ln2$

How to prove $$\sum_{k=1}^{\infty}\frac{(-1)^k}{(2k+1)^2}(2H_{2k}+H_k)\stackrel ?=\frac{\pi^3}{32}-2G\ln2,$$ where $G$ is the Catalan's constant. Attempt For the first sum, $$\sum_{k=1}^{\infty}...
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Nonlinear least squares with analytical solution

I want to find a "true" nonlinear least squares problem which does have an analytical solution. I tried to construct something with a Dirac-Delta function and ended up with $y_n = c^2\delta(x_n-x_1)...
6
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1answer
89 views

Integral $\int_a^\infty \frac{\arctan(x+b)}{x^2+c}dx$

I was playing around with some integrals and noticed that some integrals of the form: $$I(a,b,c)=\int_a^\infty \frac{\arctan(x+b)}{x^2+c}dx$$ Does have a closed form. I am trying to find for what ...
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2answers
115 views

Finding the definite integral $\int_1^e \frac{dx}{x\sqrt{1+\ln^2x}}$

So I have the following problem: $$\int_1^{e} \frac{1}{x\sqrt{1+\ln^2x}}dx $$ Can somebody comfirm that the integral of this is $$\ln|\sqrt{1+\ln^2x}+ \ln x|+C$$ and I that the anwser is $$\ln |\...
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0answers
17 views

special interior angles of regular $n$-gon

Let $K$ be a regular $n$-gon in the plane. Assume the following: $$n\in\Bbb N,\quad n\geq3\\I=\{0,1,...,n-1\}\\ \forall i\in I, \quad M(i):=\operatorname{mod}(i,n)$$ And we define $P_i$ as the $i$-th ...
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3answers
165 views

Evaluating the integral $\int_0^1 \frac{\cos bx}{\sqrt{x^2+s^2} }dx$

I'd really love to evaluate this integral exactly in terms of known functions, because for large $b$ it becomes a pain numerically. $$I(b,s)=\int_0^1 \frac{\cos bx}{\sqrt{x^2+s^2} }dx$$ Didn't get ...
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1answer
42 views

Proving the Brouwer Fixed Point Theorem

Brouwer's Fixed Point Theorem: Let $f : D^{n+1} \to D^{n+1}$ be a continuous map, then $f$ has a fixed point. The proof goes something like this: Proof: Suppose that $f : D^{n+1} \to D^{n+1}$ is a ...
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1answer
114 views

On the integral $I(a)=\int_0^1\frac{\log(a+t^2)}{1+t^2}dt$

Consider the parameter integral $$I(a)=\int_0^1\frac{\log(a+t^2)}{1+t^2}dt\tag1$$ where $\log$ denotes the natural logarithm and $a\in\mathbb{C}$. I am struggling to evaluate this integral in a ...
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1answer
43 views

exact closed form expression

The question is to find the exact closed form of the expression $(0^2+...+n^2)+(1^2+...+(n+1)^2)+...+(n^2+...+(2n)^2)$ Hint (find the exact closed form of $(n)(1^2)+(n-1)(2^2)+...+(1)(n^2)$ first). ...
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Recurrence Relations-For n∈\N0 let w(n) denote the number of 1s in the binary representation of n. For example, w(9) = 2

For $n\in\mathbb{N}_0$ let $w(n)$ denote the number of $1$s in the binary representation of $n$. For example, $w(9) = 2$, since $9$ is $1001$ in binary. Try to find a closed formula for $g(n)$ in ...
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1answer
19 views

$2n$ th partial derivative of $\frac{1}{y(1+x^2)-1}$ with respect to $x$.

I need to find the $2n$ th derivative with respect to $x$ of the function $f = \frac{1}{y(1+x^2)-1}$. I tried differentiating util a pattern was founded, but that didn't happen. I think the $x^2$ is ...
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1answer
32 views

Is there a closed-form solution for the distance to the origin from the midpoint of a line connecting a point on an arc and a point on a tangent?

I have a fixed-length line $EF$, whose endpoints run along a track consisting of two straight segments joined by an arc. I'd like to create a polar equation describing the location of the centerpoint ...
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0answers
211 views

Closed form of $\sum_{n=1}^\infty q^{- n^2} z^n$

In this question the summation goes from $-\infty$ to $\infty$ and the answer has a pretty "good" closed form. Now I came across the sum $\sum_{n=1}^\infty q^{-n^2} z^n$ where $|z|<1$ and I don't ...
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0answers
65 views

Asymptotics for $ f(n) = f(n - 1) + f( n - g(n) ) $?

Define $g(x)$ as : If $f(m) =< x < f(m+1)$ for a positive integer $m$ then $g(x) = m$. Now we define $f(n)$ for strict positive integer $n$. $$f(1) = 1 $$ $$f(2) = 3 $$ $$f(3) = 7 $$ For $n &...
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Closed form for product over Gamma function

Is there a "closed form" (with which I mean an expression not involving an indexed sum or product) for any of these four products? $$\prod_{k=1}^{n} \Gamma(\frac{x}{k*2+1})$$ $$\prod_{k=1}^{n} \Gamma(...
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1answer
48 views

A limit of combination

I want to find the closed form of the limit, \begin{align*} I(k,r):=\lim_{x\rightarrow 0}\left\{\sum\limits_{j=1}^{r+2-k} (-1)^{r+3-j-k} \binom{r-j}{k-2}\frac{1}{x^j}+\frac{1}{(1+x)^{k-1}x^{r-k+2}}\...
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3answers
64 views

What is the maximum value of $\ln(x+\ln(x+\ln(x+\cdots)))-\ln x$?

This question is somewhat similar to my last set of infinite nests (see here) but this time I would like to attain an upper bound instead of the area, as $\int_1^\infty\ln x\,dx$ does not converge. ...
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1answer
259 views

Integral $\int_0^\infty \frac{\arctan(x^2)}{x^4+x^2+1}dx$

By accident while trying to evaluate a similar integral, I think originally found here, I have taken the denominator instead of $x^4+4x^2+1$ as $x^4+x^2+1$ and stumbled into the following integral: $$...
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47 views

Prove or disprove that $\int_0^{\pi}\log^2(\cos x)~dx\stackrel{?}{=}2\int_0^{\pi/2}\log^2(\cos x)~dx$

Hence I am rather new to the concept of Parseval's Theorem I tried to apply it to integrals I am familiar with to understand the theorem in more detail. Since I recently was confronted quite often ...
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0answers
46 views

Closed form of $\sum \frac {1}{p_1.p_2.….p_r}$ where $p_r$ is the $r^{th}$ prime

Following is an infinite series quite analogous to the exponential series, but composed of prime numbers only. $$\frac {1}{2} + \frac {1}{2.3} +\frac {1}{2.3.5} + \frac {1}{2.3.5.7} + \frac {1}{2.3.5....
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2answers
47 views

Antiderivative of : $t \mapsto (1-t^2)^\lambda$

I would like to find an antiderivative of the function $$t \mapsto (1-t^2)^\lambda$$ where $\lambda \in \mathbb{R}_{>0}$ I really don't know how to proceed. One idea is to use the generalized ...
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3answers
58 views

A closed form for the sum $\sum_{i=0}^j (-1)^i \binom{j}{i} (j+c-i)^k$

Let $c\in \mathbb{Q}$ be a constant, and let $k\in \mathbb{N}$ be fixed. Is there a closed form for the sum $$ L_j := \sum_{i=0}^j (-1)^i \binom{j}{i} (j+c-i)^k, $$ for $j \leq k$ ?
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The $n$ th derivative of $\ln(xy)/(1-xy)$ with respect to $x$.

I'm looking for a closed form of \begin{align} \frac{d^n}{dx^n}\left( \frac{\ln(xy)}{1-xy} \right) \end{align} I tried using the Taylor series: \begin{align} \frac{\ln(xy)}{1-xy} = \frac{\sum_{...
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1answer
44 views

Can the binomial distribution be solved for k?

I am working on a problem where, given the binomial probability density function: $$X(p,n,k) = \left(\frac{n!}{k!(n-k)!}\right)p^k(1-p)^\left(n-k\right)$$ I need a function $Y(p,n,x) = k$ where $p, n$ ...
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0answers
36 views

Express series by a closed-form expression

I have trouble to find a closed-form expression for finite series. Before it, let's take into consideration the following very straightforward example: $$S_{n}=\frac{1}{1+r}+\frac{1}{(1+r)^2}+...+\...
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0answers
79 views

Closed form for a product of sines

From this question Evaluation of a product of sines I allready know that: $$\prod_{k=1}^{n-1} \sin(\frac{k\pi}{n}) = \frac{n}{2^{n-1}}.$$ I am interested in a closed form for the following product: $...
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2answers
64 views

Is there a closed form for $\sum_{k=0}^n2^k\binom{2n+1}{2k}$, the sum of binomial coefficients times powers of two, for even indices?

I'm hoping for a nice and simple closed form for the sum $$\sum_{k=0}^{n}2^k\binom{2n+1}{2k}.$$ Searching this site I found many nondescript titles but no duplicates, though I wouldn't be surprised if ...
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4answers
67 views

Summation of $\sum_{i=1}^n i2^i.$ [duplicate]

An intermediate step in a problem I was working on was to find a closed form for the sum $$\sum_{i=1}^n i2^i.$$ WolframAlpha returns $2^{n+1}(n-1) + 2$, but didn't provide any step-by-step solution. ...
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1answer
69 views

Integration problem related to Gamma function: $ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du $

During my work on some statistics problem, I stumbled across the following integral: $$ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du,\qquad \alpha, b, c>0 $$ I tried ...
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0answers
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Is there a closed form for $\zeta(\pi)$?

What is $\zeta(\pi)$? I know that $\operatorname{Re}(\pi)>1$, thus $$\zeta(\pi)=\sum_{n\geq1}\frac{1}{n^\pi}$$ Yet I have no idea how to even begin evaluating this series. It's probably ...
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2answers
256 views

Proving that $\sum_{n=1}^\infty \frac{\sin\left(n\frac{\pi}{3}\right)}{(2n+1)^2}=\frac{G}{\sqrt 3} -\frac{\pi^2}{24}$

Trying to show using a different approach that $\int_0^1 \frac{\sqrt x \ln x}{x^2-x+1}dx =\frac{\pi^2\sqrt 3}{9}-\frac{8}{3}G\, $ I have stumbled upon this series: $$\sum_{n=1}^\infty \frac{\sin\...
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0answers
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Evaluate $\int_0^1 \log^n(x^a)\log^m(1-x^{\color{red}{\alpha}})x^b(1-x^{\color{red}{\beta}})^t~dx$ with $\alpha\ne\beta$

Recently dealing with algebraic integrals composited out of logarithms and powers I learned about using the derivatives of the Beta Function in order to evaluate them. Applying this knowledge I was ...
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On the necessitation of $(-1)^n$ within the series expansion of $f(x)$ concerning the usage of Ramanujan's Master Theorem

Ramanujan's well known Master Theorem states that the series expansion of the transformed function $f(x)$ has to be in form of $$f(x)~=~\sum_{n=0}^{\infty}(-1)^n\frac{\phi(n)}{n!}x^n\tag1$$ ...
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2answers
81 views

Show that $\int_0^{\frac{\pi}2}4\cos^2(x)\log^2(\cos x)~dx~=~-\pi\log 2+\pi\log^2 2-\frac{\pi}2+\frac{\pi^3}{12}$

Within this collocation of definite integrals number $30.$ is given by $$\int_0^{\frac{\pi}2}4\cos^2(x)\log^2(\cos x)~dx~=~-\pi\log 2+\pi\log^2 2-\frac{\pi}2+\frac{\pi^3}{12}\tag1$$ I figured out ...
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2answers
80 views

Explicit formulas for $\operatorname{Re}(z^n)$ and $\operatorname{Im}(z^n)$

I'm looking for a closed formula for the real and imaginary part of $z^n = (u + iv)^n$. We have $$\operatorname{Re}(z^{n+1}) = u\operatorname{Re}(z^{n}) - v\operatorname{Im}(z^{n})$$ $$\...
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2answers
131 views

Integral $\int_0^1 \frac{(x^2+1)\ln(1+x)}{x^4-x^2+1}dx$

Greetings a while ago I met this integral $$I=\int_0^1 \frac{(x^2+1)\ln(1+x)}{x^4-x^2+1}dx$$ To be fair I spent some time with it and solved it in a heuristic way, I want to avoid that way so I won't ...
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3answers
62 views

Looking for closed-form solution of the following integral

I have been trying to calculate the following triple integral: $$ I(a,b,c) \,=\, \int_{x=0}^{a}\int_{y=0}^{b}\int_{z=0}^{c} \frac{dx\,dy\,dz}{(1+x^{2}+y^{2}+z^{2})^{3}} $$ I can find values ...
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2answers
46 views

Is there a closed form expression for the first zero of the first Bessel function?

$j_{1,1}$ denotes the first zero of the first Bessel function of the first kind. (That's a lot of firsts!) It's approximately equal to $3.83$. My question is, is there any closed form expression ...
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2answers
35 views

Special solution to $a+e^a\ln x = x+e^a\ln a = a+e^x\ln a$

I was messing around with the equations of the form $a+e^b\ln c$. I set two variables equal and graphed them and I noticed something that interested me enough to ask about. Let $x\in\mathbb{R}$. For ...
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1answer
34 views

How to find the closed form solution for the multivariate recurrence?

Recurrence relation: f(n,k) = f(n-1,k) + f(n-1,k-1) + f(n-2,k-1) Initial conditions: ...
3
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2answers
45 views

Closed-form expression for infinite series related to a Gaussian

Consider the following infinite series, where $x$ is indeterminate and $r$ is held constant: $\displaystyle 1 + \frac{x}{r} + \frac{x^2}{r^2} + \frac{x^3}{r^3} + ...$ It is relatively easy to see ...
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1answer
50 views

Solving (or approximating a solution of) $c = (1+a)^x - (1-a)^x$

I am looking to solve the following equation: $$(1+a)^x - (1-a)^x = c$$ I know that $c ≥ 1$, $a \in [0; 1]$, and I am looking for the only solution where $x ≥ 0$. I am aware that this is a ...
0
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1answer
52 views

closed form formula question

How do I find the closed form sum formula of $$\displaystyle\sum_{k=1}^{\log_2(n)}\log_2 k,$$ I was trying to do $\log_2 1 + \log_2 2 + \cdots + \log_2(\log_2 n)$ and using property of $\log$: $\log ...
3
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0answers
45 views

On limit of sum $\lim\limits_{s\to0^+}\sum\cos\left(\pi\frac{n}{m}\right)/{n^s}$ & $\lim\limits_{s\to0^+}\sum\sin\left(\pi\frac{n}{m}\right)/{n^s}$

$(1).$ Show that: $$ \lim_{s\to0^+}\,\left[\sum_{n=1}^{\infty}\cos\left(\pi\frac{n}{m}\right)\frac{1}{n^s}\right]=\color{red}{-\frac{1}{2}} \quad\colon\space\forall\,m\in\mathbb{N}^{+}\tag{1} $$ ...
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2answers
87 views

What is $\int_0^{\infty} e^{\frac{-(\ln(x)-\mu)^2}{2 \sigma^2}}\, dx$ [closed]

Is there a closed form solution for this? $$\int_0^{\infty} \exp\left({\frac{-(\ln(x)-\mu)^2}{2 \sigma^2}}\right)\, dx$$
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2answers
81 views

Is it possible to solve $\cos(x) + 2e^{x} = 0$ analytically?

My Calculus textbook uses $f(x) = \sin(x) + e^{2x}$ as an example of a function with infinitely many local extrema. That much is clear, because $\cos(x) + 2e^{x} =0 $ has infinitely many solutions ...
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0answers
61 views

Closed form from recurrence

We have $a(0)=a(1)=1$, $$a(n)=(-1)^{n-1}+2\sum\limits_{k=1}^{n-1}\binom{n}{k}(-1)^{n-k-1}a(k)$$ $$1,1,3,13,75,541,4683,\cdots$$ which has nice closed forms, ex. $$a(n)=\sum\limits_{k=0}^{n}k!{n\brace ...
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0answers
13 views

Is there any closed form exists for the two tables?

I am working on a combinatorial problem and have the following observations represented in the form of two tables. Is there any known patterns or closed form exists for either of the two tables that ...
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0answers
53 views

Closed form for $0^n0!+1^n1!+\cdots+k^nk!$

We have $$\sum\limits_{k=0}^{m}k^nk!$$ Is there a nice closed form for this sum?