Reciprocals in real analysis

Implicit definition

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

By power series

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

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

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

1x{lim a0 (a) n=0 a n(x+f(a+1)) n x(,0) lim a0 +a n=0 (a) n(x+g(a1)) n x(0,) \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.

Reciprocals in complex analysis

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.