Talk:Sum of Complex Indices of Real Number

You're right, my bad, I do have to address the $2k\pi i$. I'll invoke the periodicity of the exponential. Or does the thm onky hold on the principal branch? I'm on my phone right now so I can't correct it until later, I need to think about this a bit. --GFauxPas (talk) 17:00, 4 December 2016 (EST)

Offhand I'm not sure but I know that the multi-value nature of $r^z$ is infinitely more complicated than you think.

I expect you can demonstrate that the elements of the multi-set that is $r^{\phi + \psi}$ are precisely those that you get when you multiply all the elements of the multi-set that is $r^{\phi}$ by all the elements of the multi-set that is $r^{\psi}$, but I'm not placing any money on it at this stage of the weekend.

Hint: analyse $1^z$ for the general: $z \in \Z$, $z \in \Q$, $z \in \R$, $z \in i \R$, $z \in \C$ and note each level of complexity as you go. Now you understand why I never really tackled complex analysis, the fundamentals take a lot of work. --prime mover (talk) 17:11, 4 December 2016 (EST)

You're right. I think I'll cop out and just restrict the theorem to the principal branch. It's all I need for the time being. --GFauxPas (talk) 18:32, 4 December 2016 (EST)

Even that might not work, because the product of 2 numbers in the principal branch may not itself be in the principal branch. --prime mover (talk) 01:11, 5 December 2016 (EST)

But for the principal branch, $\log r = \ln r + i \arg t$ and $\theta = 0$ because $t>0$ is real, no? I realize we need a definition for the principal branch of $t^\psi$, you're right, this is a lot of work. I'm doing all this just to have the Laplace Transform of Power! --GFauxPas (talk) 06:59, 5 December 2016 (EST)

At that stage I would go and consult a textbook. :-) --prime mover (talk) 07:07, 5 December 2016 (EST)