Definition talk:Spence's Function

Re {{explain}}

Made a bit of a mess of this - but my motivation for using the notation $\int_0^z$ and choosing a straight line path from $0$ to $z$ was to handle both real and complex arguments in a common definition (for real $z$ you're just following the real axis) specifically, rather than split it into the real case and the complex case, which I think should be generally avoided to make maintenance easier. (similarly this was my motivation for using $\operatorname{Ln}$ to avoid complications with the complication of multiple values and so forth) A few definitions are a bit messy in this regard. The way that it's been defined here is common with literature I've found, but maybe the authors didn't pay much attention to the technicalities either, so I'm unsure whether it's abuse of notation. Input on that would be appreciated. Proving analyticity of the principal log might be useful (and a proof/derivation of the region in which this is true) - and if not covered elsewhere it'd be useful to prove that contour integrals of analytic/holomorphic functions on simply connected regions are independent of path. I'll definitely get this definition sorted (and maybe look into making proofs of the aforementioned facts) before I post any more theorems related to the function, sorry if I'm a bother. Caliburn (talk) 13:57, 5 March 2018 (EST)

We (collectively, as a websiteful of contributors) have been wrangling over how we should define mathematical functions which are "the same, but different" over the real and complex domains. My preferred technique is to set up a master page and transclude the real and complex definitions as subpages, but some contributors believe this is overkill, and that we ought to just have the one. Unfortunately the subtleties between the domains (i.e. the reals can be ordered, and are one-dimensional, and the complex are two-dimensional and can't) means that the treatment afforded the real case cannot (usually) be directly applied to the complex case.

We also have to be careful about the fact that in most cases we have not got round to rigorously defining many of the functions in the complex plane in the first place, let alone demonstrated that the real case is a genuinely a restriction of the complex case (I did make a start in it some years back but got sidetracked by something else), and indeed, much of the foundational underpinning of complex analysis has still not properly been completed (although Anghel did a lot of good work there as well).

Hence our reluctance to commit to a "one page fits all" approach. If nothing else, "explain" templates are just a flag to alert us to the fact that there still stuff to be done. There are plenty of such areas waiting for someone to fill it all in. --prime mover (talk) 14:35, 5 March 2018 (EST)