General Mathematics

   

Tutorial: The Real Powers from Extension of Two Determining Properties of Positive-Integer Powers

Authors: Steven Kenneth Kauffmann

This tutorial parlays two determining properties of positive-integer power functions of a positive real variable into closed formulas for the real power functions of a positive real variable. These two determining properties are that any positive-integer power of unity equals unity, and the linear first-order differential equation that a positive-integer power function of a real positive variable satisfies, which is implicit in its derivative. These two determining properties of positive-integer power functions of a positive real variable are extended to arbitrary real values of the positive-integer power. The extended linear first-order differential equations and initial conditions are then used to generate the Taylor expansions of those real power functions of a positive real variable around the zero value of the real power; this can be carried out in at least two different ways. Those Taylor expansions converge for every real value of the power and every positive real value of the variable, and are readily reexpressed entirely in terms of the exponential function and its inverse; one thus has closed formulas for all the real power functions of a positive real variable. Logarithms describe arbitrary positive numbers as real powers of a given positive number; they can expressed entirely in terms of the exponential function's inverse. The value of the particular positive constant whose powers yield the exponential function itself is worked out.

Comments: 5 Pages.

Download: PDF

Submission history

[v1] 2019-09-22 08:27:44
[v2] 2019-09-26 06:39:58 (removed)
[v3] 2019-09-27 20:16:27 (removed)
[v4] 2019-09-29 20:36:38

Unique-IP document downloads: 19 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus