Talk:Sum of Complex Numbers in Exponential Form/General Result

With regards to the handwaving comment, which for some unknown reason isn't rendered when I look at the page, "Strictly speaking we need to consider the case where any individual $z_k = 0$ as otherwise the initial statement of the theorem is flawed -- in such a case there is no $\theta_k$ in the first place.":


 * Whoops my bad, fixed comment. (Exercise for student to work out what the problem was and why what I did fixed it.) --prime mover (talk) 02:14, 31 December 2019 (EST)

I think the simplest solution is to add the restriction that $r_k > 0 \, \forall k$ in the theorem as it still retains the intent of the generalized result.

One might argue that we should add more rigor to the handling of edge cases, e.g. any or all of the $r_k = 0$, to demonstrate "how to do it rigorously"; however, my largest stumbling block was the partial definition in Argument of Complex Number as it doesn't mention how to deal with the case where the modulus is $0$. I've always been taught that $\arg 0 = undefined$; maybe that should be added to that Definition page? Most texts I've seen skirt the issue by stating that $z = x + i y = r \paren { \cos \theta + i \sin \theta }$ and then state that $r$ is the modulus and $\theta$ is the argument. https://en.wikipedia.org/wiki/Argument_(complex_analysis)#Computing_from_the_real_and_imaginary_part does at least define it to be $undefined$ for $0$.

It's almost like the "choice" we have to use the Axiom of Choice where we need to choose between rigorously permitting the summation of one or more $\map \sin {undefined}$ and $\map \cos {undefined}$ or simply not allowing such a summation by requiring $r_k > 0 \, \forall k$.


 * All the above is way over my head. Someone else is going to have to consider it. --prime mover (talk) 02:14, 31 December 2019 (EST)

P.S. Thank you User:Prime.mover for sorting-out how to format those equations - at least I now have an example I can refer to. --John Coupe (talk) 18:50, 30 December 2019 (EST)


 * The page Definition:Polar Form of Complex Number defines the polar form (and consequently the exponential form) of $0+0i$ with $\theta = 0$. Obviously this is arbitrary and not an intentional definition, but we could consistently use it.


 * However some of the theorems referred to also seem to use the $\arg$ function without considering $z = 0$, so I think a more thorough review would be in order.


 * For this theorem the easy way out is indeed to require $z_1 \ldots z_k$ nonzero. I'll make the adjustment. &mdash; Lord_Farin (talk) 07:54, 31 December 2019 (EST)