Talk:Exponent Combination Laws

Rational Power Subpage

I would like to create a subpage, showing these results hold for rational powers (working from the ground up, obviously). It seems that the general theme on proofwiki would be to split this page into two subpages: "Exponent Combination Laws\Real Powers" and "Exponent Combination Laws\Rational Powers", and then transcluding the statements of both subpages into the parent page. However, that would mean the repetition of 7 statements. What's the best way to do this, without having so much repetition? --Keith.U 00:39, 15 June 2016 (UTC)

I would advise against this level of complexity. If it proves necessary to separate individual pages into real and rational versions, then do those individual pages. This is a convenient summary page that reports on the overall results. Once it's proved for the real version, it's automatically true for the rational version.

And the only reason you may need to separate these out for the rational versions is because you need to prove the rational versions as a stepping-stone for the real versions -- or because they are used to establish the definition of the real version. --prime mover (talk) 05:13, 15 June 2016 (UTC)

The second point is precisely why I'm interested in proving these properties. I think it's interesting to show that, no matter which popular definition of the exponential we take, we can prove all of the important properties, in the spirit of that definition. (E.g., using the machinery of power series to prove properties under the series definition, or using the machinery of ODE's to prove properties under the ODE definition.) The definition of $\exp$ as an extension of rational powers is particularly important, in my opinion at least, because I think it is the most intuitive of the definitions. However, this route is so rarely taken (likely due to how tedious such a development is).

Obviously, this not necessary. I do not mean to flood the site with a torrent of uninteresting results and laborious proofs just to satisfy a "project" that is interesting only to me. If you think such a development is not within the nature of proofwiki, I'll default to your judgement. --Keith.U 08:03, 15 June 2016 (UTC)

No problem, go ahead with this -- but please don't duplicate this "Exponent Combination Laws" page with a rational version. The individual proofs of the individual results, by all means. --prime mover (talk) 17:20, 15 June 2016 (UTC)

Changed my mind -- it seems like a good idea after all. --prime mover (talk) 20:52, 28 June 2016 (UTC)

Complex Powers

I have need of this result for $t^{a+bi} = t^a t^{bi}$. However, I'm not sure where to put it. An extension of this page's result would call for $t^{\psi + \tau}, \psi,\tau \in \C$. The result I need is weaker, but my intuition is that it gets its own page, rather than being a corollary of this page. Thoughts? --GFauxPas (talk) 08:46, 29 November 2016 (EST)

The main result $t^{\psi + \tau}$ for complex $\psi, \tau$ should be the main result and then $t^{a+bi} = t^a t^{bi}$ is a consequence. --prime mover (talk) 12:31, 29 November 2016 (EST)