What is Closed-form solution: Definition and 12 Discussions

In mathematics, a closed-form expression is a mathematical expression expressed using a finite number of standard operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration. The set of operations and functions admitted in a closed-form expression may vary with author and context.

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  1. tworitdash

    MATLAB Closed form not the same as the discrete form

    clear; lambda = 3e-2; x = 4 * pi/lambda * linspace(eps, 15, 100000); T = 5e-3; t = [0:0.001e-3:T] ; % 0.1:1e-3:0.1+T]; u = 3; a = 4*pi/lambda * u; for i = 1:length(x) Z(i) = sum(-((cos(a.*t) - cos(x(i).*t)).^2 + (sin(a.*t) - sin(x(i).*t)).^2)); end % Z1 = csc((a+x)/2) .*...
  2. hilbert2

    A Anharmonic oscillators with closed-form solutions

    There are some articles from the 1980s where the authors discuss 1D quantum oscillators where ##V(x)## has higher than quadratic terms in it but an exact solution can still be found. One example is in this link: https://iopscience.iop.org/article/10.1088/0305-4470/14/9/001 Has anyone tried to...
  3. stevendaryl

    I Closed-Form Solution Desired for Infinite Series

    Is there a simple closed-form solution for the following infinite series? ##F(a,b,c) = \sum_{j=0}^\infty \frac{(j+a)!}{(j+b)! (j+c)!}## where ##a, b, c## are positive integers?
  4. E

    I Closed-form solution for a triple integral

    Hello all, I need to evaluate the following 3-dimensional integral in closed-form (if possible) \int_{y_1=0}^{\infty}\int_{y_2=0}^{\infty}\int_{x_2=0}^{zy_2}\exp\left(-\min(x_2,\,y_1(z-\frac{x_2}{y_2}))\right)e^{-(K-1)x_2}e^{-y_1}e^{-y_2}\,dx_2dy_2dy_1 where ##z## is real positive number, and...
  5. E

    I Closed-form solution for an integral

    Hello all, Is there a closed form solution for the following integral \int_0^z\frac{1}{1+z-x}\frac{1}{(1+x)^K}\,dx for a positive integer ##K\geq 1##, and ##z\geq 0##? I searched the table of integrals, but couldn't find something similar. Thanks in advance for any hint
  6. kevin2016

    A What is the closed-form solution using ALS algorithm to optimize

    C \in \mathbb{R}^{m \times n}, X \in \mathbb{R}^{m \times n}, W \in \mathbb{R}^{m \times k}, H \in \mathbb{R}^{n \times k}, S \in \mathbb{R}^{m \times m}, P \in \mathbb{R}^{n \times n} ##{S}## and ##{P}## are similarity matrices (symmetric). ##\lambda##, ##\alpha## and ##\beta## are...
  7. tom.stoer

    Closed-Form Solution for f(A,B) Equation

    Does anybody know a closed for f(A,B) = \sum_{n\in\mathbb{Z}}e^{-Bn^2 + iAn} with A real and B positive.
  8. L

    Closed-form solution of a quadratic optimization problem

    Hello, My question is as follows. Is it possible to obtain a closed form solution to \displaystyle \max_{\xi\ge 0, \lambda\ge 0}\,\, -\frac{1}{2}\||\xi\||^2 +(\xi,\,\lambda) -\frac{1}{2}\||\lambda\||^2 Here \xi and \lambda are vectors. Thank you.
  9. J

    Is there a closed-form solution r(t) regarding universal law of gravitation?

    Newton's universal law of gravitation: F=-G \frac{m_1 m_2}{r^2} I'd like to set up the problem so the particle begins at t=0 at radius r=r0 and radial velocity vr=v0. And there is only a component of velocity, in the radial direction. (The particle is going straight toward the...
  10. bcrowell

    *simple* example of mechanics problem with no closed-form solution?

    I'm looking for a good example for a freshman mechanics class to demonstrate how one can integrate the equations of motion numerically when there is no closed-form solution. The problem below is the best I've been able to come up with yet, but I'm not totally happy with it, and I'm wondering if...
  11. J

    Establishing linearity in laplace's equation with no closed-form solution

    I'm writing a paper, and as a motivation to the forthcoming finite element modeling, I want to state, with some sort of "proof" that Laplace's equation in a heterogeneous volume: \del (sigma \del V) = 0 exhibits linearity. By "linearity", I mean that if a set of initial conditions...
  12. T

    Closed-Form Solution for f(n) w/o Generating Fns

    f(n) = 0, n \leq 2 f(n) = \sqrt{n}f(\sqrt{n}) + n, n > 2 How can I get this in closed form? Generating functions won't work. Recuring a number of times hasn't worked out for me. Or can I show that f = O(n*lg(lg(n))), where lg stands for ln(n)/ln(2), without f being in closed form? Sorry for...