Definition talk:Exponential Function/Complex

I'm still dissatisfied with the way this all hangs together. Granted we have several definitions of the exponential in the real domain, which is where it originates. Then we have another definition for the complex plane, and it's the same as for the real domain, except it's an extension to complex numbers. So it ought to be possible just to have one set of definitions, and have it for the complex plane, and then point out that "in the real plane, it's the same, but restricted to real numbers" or whatever.

At the moment we've got reals, divided into 5 definitions, and complex, divided into 4 definitions. I wonder whether it would fit better (and require less duplication) to divide it primarily into the definitions, and secondarily, within those definitions, into complex and real. Thus you'd have "as a limit of a sequence", which is the same for both real and complex; "As a sum of a series" again, same for real and complex, etc., and then there are the definitions which are specific to one or the other.

Thus, when we get round to e.g. defining the exponent of a square matrix $\mathbf A$ (it comes up in physics, Lie group theory IIRC) we can "just" extend the individual relevant definition (which is the "sum of series" definition: $\displaystyle e^{\mathbf A} = \sum_{n \mathop \in \N} \frac {\mathbf A^n} {n!}$ etc.).

It may also be appropriate to design the individual pages so as to be able to transclude them into both "definition of exponential for real numbers" and "definition of exponential for complex numbers" so as to allow the same page to be written once and transcluded twice

Yes I understand there's a lot of work to be done so as to achieve this - and it will still need some significant analysis work to get there in the first place - not to mention the relevant adjustment of the links. So it can't be done without a lot of thought and preparation. It's not urgent - but would make an interesting and useful long-term project for anyone who cares to take it up. --prime mover (talk) 21:27, 27 January 2013 (UTC)


 * I agree that there's definitely room for improvement, so feel free so put the "refactor" template back up again. It's a good idea to put the similar definitions on the same page. I'm not so certain about the idea of designing the pages to be transcluded twice, because it's common to write real variabels as $x$ and complex variabels as $z$. I know that there's other contributors who have written about the complex exponential - what do they think? --Anghel (talk) 22:47, 27 January 2013 (UTC)


 * My vote: Defs together, no double transclusion - the latter would have undesirable consequences on the long run, or so I predict. --Lord_Farin (talk) 09:59, 28 January 2013 (UTC)

Codomain
Regarding:


 * The codomain of $\exp$ is $\C \setminus \left\{ {0}\right\}$, as shown in Image of Complex Exponential Function.

I don't think it's necessary to prove that $\C \setminus \left\{ {0}\right\}$ is the codomain. It needs to be proven that $\C \setminus \left\{ {0}\right\}$ is the image, but to me that should be in an "also see". --GFauxPas (talk) 21:11, 28 January 2013 (UTC)


 * Agreed. --Lord_Farin (talk) 21:17, 28 January 2013 (UTC)


 * Good catch - I missed that. Codomain can of course be whatever you want it to be as long as it's a superset of the image. In this context it "makes sense" for it to be $\C$, although I can also see an arguable case for it to be $\C \setminus \left\{{0}\right\}$. However, it does make it conceptually easier to define it as $\exp: \C \to \C$. --prime mover (talk) 21:23, 28 January 2013 (UTC)