# 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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### Formula for one element in the intersection of two cosets $m\Bbb{Z} + a$ and $n \Bbb{Z} + b$ whenever $\gcd(m,n) = 1$ is true. [duplicate]

We know that $(m\Bbb{Z} + a) \cap (n \Bbb{Z} + b) = \text{lcm}(m,n) \Bbb{Z} + x$, that is if the intersection is not empty, where $x$ is any element of the latter coset. Is there a formula for $x$ in ...
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### L21 norm closed form

Please help me Let X, A, B, and C are 3rd order tensor and a and b are constant. I need the closed form for this problem min_A a*||A||_2,1 +b/2* ||A-(X-B+C/b)||_F^2
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### How to solve exposure gain compensation equation?

I am working on panorama stitching and more specifically exposure compensation between images. I am trying to understand how to solve equation (29) from Automatic Panoramic Image Stitching using ...
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### How to solve a recursive equation with fraction structure

Consider the following recursive equation: $$a(K) = K \frac{a(K-1) + b}{a(K-1) + b + S},K=1, 2, \cdots$$ where $b \in \{1, 2, \cdots \}$ and $S>0$ are both constants. Is there any way to solve ...
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### Ideas for a Closed form for $\sum_{k=0}^n k10^k$

Is there a closed formula for this summation: $$\sum_{k=0}^n k10^k,$$ where $n\in\mathbb{N}$? I would like to learn trick o strategies for this kind of problems.
1answer
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### What general kinds of closed-form problems are there?

A closed-form expression is a mathematical expression that contains only finite numbers of symbols and operations from a given set. A mathematical problem is a closed-form problem if its solution is ...
1answer
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### Understanding Wilfian formula?

In this paper by Igor Pak, he mentions the following definition of wilfian formulas, which basically aim at characterising what amounts to a good enumeration formula, the classification goes as ...
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### How to express $n$ unknown variables in a system with $n-1$ equations

I have a system of $n-1$ equations which have $n$ unknown variables, as follows: \begin{align} &\text{Equation $1$} &&x_1A_{1,1}+...+x_nA_{1,n}=0\\ & \hspace{1cm}\vdots && \...
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### Computing $\int_0^1 \frac{(1-t)(1+at)}{|1+at|^p} dt$.

Just as the title suggests, I want to compute the integral of the form $$\int_0^1 \frac{(1-t)(1+at)}{|1+at|^p} dt$$ in terms of $a$ and $p$, where $a \in \Bbb R$ and $p\in (1,2)$. This is not a ...
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### Inverse is not connected

Let $f:[a, b] \rightarrow[a, b]$ be continuous. Provide a counterexample for the following statement: The inverse image $f^{-1}([c, d])$ is connected for any $[c, d] \subset[a, b]$. One of the ideas ...
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### Incomplete upper gamma for a non-integer number of degrees of freedom

I can't seem to nail the closed form of the incomplete upper gamma function for the number of degrees of freedom $s$ being a fraction type $n/2$ where $n$ is integer. For the case when $s$ is integer ...
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### Closed form of $\sum_{n=1}^{\infty} \frac{a^n}{n(n+1)}$

I have come across the series$$\sum_{n=1}^{\infty} \frac{a^n}{n(n+1)}, \quad|a|<1$$ in an intermediate step while solving a problem. I wonder if there is a closed form for it? I could only observe ...
1answer
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### Prove $\int_{0}^{\pi}\frac{u}{1-\cos u}\ln\frac{1+\sin u}{1-\sin u}{d}u=\left(\pi+2\ln2\right)\pi$ [closed]

How to prove $$\displaystyle\int_{0}^{\pi}\frac{u}{1-\cos u}\ln\left(\frac{1+\sin u}{1-\sin u}\right)\mathrm{d}u=\left(\pi+2\ln2\right)\pi\,\,?$$ I tried to apply the Feynman method to get the ...
1answer
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### Closed form expression for eigenvectors of 2x2 matrix?

Is there a closed form expression for the eigenvalues/eigenvectors of an arbitrary 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$? Wolfram|Alpha tries to provide an ...
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### Closed form for a minimum related to Khintchine inequality

Let $p>2$ be a real number. In this blog post by George Lowther, a proof of the right-hand Khintchine inequality is given where $$m_p:=\min_{x>0}\;x^{-p}\cosh(x)$$ comes into play. Question: Is ...
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### Closed form for definite integral involving Bessel function, $K_1$

Wolfram Alpha knows that, and it can be calculated that: $$\int_0^1 \exp\bigg(\frac{1}{\log x}\bigg)~dx=2K_1(2).$$ Where $K_1$ is a modified Bessel function of the second kind. I wanted to find out ...
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### Finding open / close form from this recurrence relation [closed]

In Nlogonia, buses and minibuses are 18 and 12 meters long, respectively. Buses can only be green; the minibuses can be blue or red. Two vehicles of the same color are indistinguishable. One of the ...
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