[[!redirects Sandbox > history]] [[!redirects Sandbox]] < [[nlab:Sandbox]] We need Archimedean ordered fields which has all analytic functions... What is an analytic function? * Wikipedia, [Analytic function](https://en.wikipedia.org/wiki/Analytic_function) Given real number $a$ and positive real number $\epsilon$, let $(a - \epsilon, a + \epsilon) \subseteq \mathbb{R}$ denote the open interval with endpoints $a$ and $b$. An analytic function is a function $f:(a - \epsilon, a + \epsilon) \to \mathbb{R}$ with a sequence $c:\mathbb{N} \to \mathbb{R}$ such that for all real numbers $x$ such that $a - \epsilon \lt x \lt a + \epsilon$, $f(x)$ is the limit of the power series $$\sum_{i = 0}^{\infty} c(n) (x - a)^n$$ We just add to the Archimedean ordered field the axiom that it has all analytic functions. We need analytic continuations to continue the functions to the unbounded situation. * Wikipedia, [Analytic continuation](https://en.wikipedia.org/wiki/Analytic_continuation) which is necessary for defining the commonly used analytic functions, such as the exponential, logarithm, trigonometric functions and their inverses. category: redirected to nlab