reciprocal

**analysis** (differential/integral calculus, functional analysis, topology)

metric space, normed vector space

open ball, open subset, neighbourhood

convergence, limit of a sequence

compactness, sequential compactness

continuous metric space valued function on compact metric space is uniformly continuous

…

…

In real analysis, the reciprocal $\frac{1}{x}$ is a partial function implicitly defined over the non-zero real numbers by the equation $x \left(\frac{1}{x}\right) = 1$. This is the definition commonly used when defining the real numbers as a field.

Let us define the functions $f:(-1,1)\to\mathbb{R}$ and $g:(-1,1)\to\mathbb{R}$ from the open subinterval of the real numbers $(-1,1) \subset \mathbb{R}$ to the real numbers $\mathbb{R}$ as the locally convergent power series

$f(x)\coloneqq -\sum_{n=0}^{\infty} x^n$

$g(x)\coloneqq \sum_{n=0}^{\infty} (-1)^n x^n$

The **reciprocal** $\frac{1}{(-)}:(-\infty,0)\union(0,\infty)\to\mathbb{R}$ is then piecewise defined as

$\frac{1}{x} \coloneqq
\begin{cases}
\lim_{a\to 0^-} (-a) \sum_{n=0}^{\infty} a^n (x+f(a+1))^n & x \in (-\infty,0) \\
\lim_{a\to 0^+} a \sum_{n=0}^{\infty} (-a)^n (x+g(a-1))^n & x \in (0,\infty)
\end{cases}$

This definition implies that the reciprocal is analytic in each of the two connected components of the domain.

The reciprocal in complex analysis should be the analytic continuation of the reciprocal in real analysis.

Last revised on June 4, 2021 at 20:43:54. See the history of this page for a list of all contributions to it.