nLab real nth root function

Context

Algebra

Analysis

Contents

Idea

Given an integer nn, The different types of nn-th root partial functions on the real numbers that satisfy the functional equation f(x) n=xf(x)^n = x on some subset of the real numbers.

Definition

Let us define the real numbers to be a sequentially Cauchy complete Archimedean ordered field, since that is the minimum requirement for which the inverse function theorem is true.

There exists an nn-th root function for every integer nn \in \mathbb{Z} apart from zero, defined on the positive real numbers by the equation:

0<|n|xnexp(1nln(x))0 \lt \vert n \vert \vdash \sqrt[n]{x} \coloneqq \exp(\frac{1}{n} \ln(x))

Now, assume that n>1n \gt 1. Then one could extend the nn-th root function to a function defined on the half-open interval [0,)[0, \infty). Let us define continuous functions f k:[0,)[0,)f_k:[0, \infty) \to [0, \infty):

f k(x)={1/n k whenx1/n 2k exp(1nln(x)) whenx1/n 2k f_k(x) = \begin{cases} 1/n^k & \mathrm{when}\; x \leq 1/n^{2 k} \\ \exp(\frac{1}{n} \ln(x)) & \mathrm{when}\; x \geq 1/n^{2 k} \\ \end{cases}

As stated, that requires knowing whether x1/n 2kx \leq 1/n^{2 k} or x1/n 2kx \geq 1/n^{2 k}, but it is possible to work around this by patching three functions together:

  • f k 0:[0,1/n 2k)f^{0}_k:[0,1/n^{2 k}) \to \mathbb{R} is the linear function f(x)=n kxf(x) = n^k x,
  • f k +:(0,)f^{+}_k:(0,\infty) \to \mathbb{R} is defined as min(n kx,exp(1nln(x)))\min(n^k x,\exp(\frac{1}{n} \ln(x))).

Since these functions agree on their overlap, and their domains comprise all of [0,)[0, \infty) we do get a total function f k:[0,)[0,)f_k:[0, \infty) \to [0, \infty) as a result.

Now the sequence of functions (f k) k=0 (f_k)_{k=0}^\infty so defined converges uniformly on any bounded interval to a continuous function ()n:[0,)[0,)\sqrt[n]{(-)}:[0, \infty) \to [0, \infty) called the principal nth root function. It is easily seen that (xn) n=x(\sqrt[n]{x})^n = x and x nn=x\sqrt[n]{x^n} = x.

Additionally, if nn is an odd integer greater than 1, then there exist nn-th root functions defined on the entirety of the real numbers.

Let us patch this sequence of triples of functions together:

  • f k :(,0)f^{-}_k:(-\infty,0) \to \mathbb{R} is defined as max(n kx,exp(1nln(x)))\max(-n^k x,-\exp(\frac{1}{n} \ln(-x)))
  • f k 0:(1/n 2k,1/n 2k)f^{0}_k:(-1/n^{2 k},1/n^{2 k}) \to \mathbb{R} is the linear function f(x)=n kxf(x) = n^k x,
  • f k +:(0,)f^{+}_k:(0,\infty) \to \mathbb{R} is defined as min(n kx,exp(1nln(x)))\min(n^k x,\exp(\frac{1}{n} \ln(x))).

Since these functions agree on their overlap, and their domains comprise all of \mathbb{R} we do get a total function f k:f_k:\mathbb{R} \to \mathbb{R} as a result.

Now the sequence of functions (f k) k=0 (f_k)_{k=0}^\infty so defined converges uniformly on any bounded interval to a continuous function ()n:\sqrt[n]{(-)}:\mathbb{R} \to \mathbb{R} called the nth root function for odd integer nn. It is easily seen that (xn) n=x(\sqrt[n]{x})^n = x and x nn=x\sqrt[n]{x^n} = x.

Other nth root functions

According to (Richman 2010), given the existence of an nth root function, there are an uncountable number of functions that satisfy the functional equation f(x) n=xf(x)^n = x on some subset of the real numbers. Each of these could be called a real “nth root function”.

For example, let 1 :1_{\mathbb{Q}}:\mathbb{R} \to \mathbb{R} be the constructive Dirichlet indicator function, defined as 1 (x)11_{\mathbb{Q}}(x) \coloneqq 1 for every rational number qq \in \mathbb{Q}, and 1 (x)01_{\mathbb{Q}}(x) \coloneqq 0 for every real number xx apart from every rational number qq \in \mathbb{Q}

q.|xq|>0\forall q \in \mathbb{Q}. \vert x - q \vert \gt 0

Then the function f(x)(1) 1 (x)xnf(x) \coloneqq (-1)^{1_{\mathbb{Q}}(x)} \sqrt[n]{x} is a real nth root function, even though it is nowhere continuous, and not defined on the entire open interval (n,)(n, \infty).

See also

References

Created on December 13, 2022 at 13:31:39. See the history of this page for a list of all contributions to it.