"Analytical" solutions are exact and obtained by methods of symbolic manipulation, derived using analysis. The methods of numerical analysis are themselves derived using (symbolic) analysis. "Analytical" really fails to convey the intended distinction for me, since both approaches seem analytical. $\endgroup$ –

2020年3月24日 · "Analytical" and "semi-analytical" are not really technical terms, I think. Roughly speaking, "analytical" means "closed-form", i.e. expressed as a finite expression using "well-known" functions. What exactly is a "well-known" function may be debated: $\sin$ and $\cos$ certainly qualify, $\text{LambertW}$ and generalized hypergeometrics maybe ...

What the more precise definition is that: In the analytical approach, you manipulate the equation algebraically to find the value of the independent variable that you desired to look for. In lower-level algebra and lower-intermediate calculus analytical approach generally gives you the most accurate answer.

2022年8月15日 · It is the same story with $\cosh$: we can use its analytical characterisation to give a rigorous definition of an angle in hyperbolic space. And the motivation behind doing this comes from the fact that earlier, we used an informal geometric definition of $\cosh$ to "derive" the analytical definition.

2020年10月14日 · I am having difficulty conceptualizing the real analysis definition of "closed" for the case of a set having no limit points. In Understanding Analysis, Stephen Abbott defines a limit poi...

2016年4月28日 · The definition in my book just says: "A function f(x) which is represented by a power series with a positive radius of convergence is said to be 'real analytic at the origin', or simply 'analytic'." To me it seems like this definition ends prematurely. Like when I was reading it I expected there to be an "if" and then a list of qualifications.

Because while the definition "differentiable in a neighborhood" is a suitably-equivalent way of saying "analytic" in the context of complex analysis, it is not necessarily the most informative way.

2024年1月23日 · Functional Analytical definition of no arbitrage. Ask Question Asked 4 months ago. Modified 4 months ago.

2020年4月19日 · I am having difficulty understanding a proof that my instructor in real analysis has written for the limit of the sequence $\\left(\\frac{4n+1}{n+3}\\right)$. She claims that the sequence converges t...

2020年1月9日 · In baby Rudin, the sine function is defined as $\frac{1}{2i} \left[\exp (ix) - \exp (-ix)\right]$, and the cosine function is defined as $\frac{1}{2}\exp (ix) + \frac ...