[[!redirects Sandbox > history]] [[!redirects Sandbox]] < [[nlab:Sandbox]] There is a hierarchy of ordered field structures Rational numbers -> Archimedean ordered fields -> Archimedean Euclidean fields -> Archimedean analytic Euclidean fields -> Cauchy complete Archimedean ordered fields -> the terminal Archimedean ordered field ---- We need Archimedean ordered fields which has all analytic functions... ## Entire functions What is an entire function? * Wikipedia, [Entire function](https://en.wikipedia.org/wiki/Entire_function) Given real number $a$, an entire function is a function $f:\mathbb{R} \to \mathbb{R}$ with a sequence $c:\mathbb{N} \to \mathbb{R}$ such that for all real numbers $x$, $f(x)$ is the limit of the power series $$\sum_{i = 0}^{\infty} c(n) \frac{(x - a)^n}{n!}$$ What is the limit of the power series? It is the limit of the sequence of partial sums $g:\mathbb{R} \times \mathbb{N} \to \mathbb{R}$ defined as $$g(x, m) = \sum_{i = 0}^{m} c(n) \frac{(x - a)^n}{n!}$$ Now we need to express it in $\epsilon$-$\delta$ terms. However, in order to do that we additionally need to assume a modulus of convergence for the series. $$\prod_{x:\mathbb{R}} \prod_{\epsilon:\mathbb{Q}_+} \sum_{N:\mathbb{N}} \prod_{m:\mathbb{N}} \prod_{m:\mathbb{N}} (m \geq N) \times (n \geq N) \to (-\epsilon \lt g(x, m) - g(x, n) \lt \epsilon)$$ $$\sum_{M:\mathbb{R} \times \mathbb{Q}_+ \to \mathbb{N}} \prod_{x:\mathbb{R}} \prod_{\epsilon:\mathbb{Q}_+} \prod_{m:\mathbb{N}} \prod_{m:\mathbb{N}} (m \geq M(x, \epsilon)) \times (n \geq M(x, \epsilon)) \to (-\epsilon \lt g(x, m) - g(x, n) \lt \epsilon)$$ Then to say that $f$ is the limit of the sequence we have $$\prod_{x:\mathbb{R}} \prod_{\epsilon:\mathbb{Q}_+} \sum_{N:\mathbb{N}} \prod_{m:\mathbb{N}} (m \geq N) \to (-\epsilon \lt g(x, m) - f(x) \lt \epsilon)$$ We talk about convergent power series. Every convergent power series has a limit function on the real numbers. ## 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 $$\sum_{i = 0}^{\infty} c(n) \frac{(x - a)^n}{n!}$$ What is the limit of the power series? It is the limit of the sequence of partial sums $f:(a - \epsilon, a + \epsilon) \times \mathbb{N} \to \mathbb{R}$ defined as $$f(x, m) = \sum_{i = 0}^{m} c(n) \frac{(x - a)^n}{n!}$$ Now we need to express it in $\epsilon$-$\delta$ terms. However, in order to do that we additionally need to assume a modulus of convergence for the series. ## Analytic continuation 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