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.
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...
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?
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...
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
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...
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.
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...
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...
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...
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...