Talk:Laplace Transform of Complex Power

What does it mean for the function to be continuous on the complex plane, when it is only defined for positive reals?

Please sign your posts. --prime mover (talk) 22:03, 2 October 2022 (UTC)

Yes, that must have been a mistake. --Usagiop (talk) 17:19, 2 October 2022 (UTC)

I corrected it so that it is at least correct. But the branch must be explained, more clearly. --Usagiop (talk) 17:33, 2 October 2022 (UTC)

Can someone check the source? I guess it was originally only discussed on the principal branch.

On the other hand, the expressions $t^q$, $s^q$ and $s^{q+1}$ in $\mathsf{Pr} \infty \mathsf{fWiki}$ are defined as multifunctions. Thus we need to mention the branches everywhere. Now it seems silently assumed that $t^q$ and $s^q$ are the same fixed branch and $s^{q+1} := s \cdot s^q$. --Usagiop (talk) 19:45, 2 October 2022 (UTC)

No, $s^q$ is not the same branch as $t^q$ in the proof. We need to assume the principal. --Usagiop (talk) 19:50, 2 October 2022 (UTC)

Note here is the case $t^q |_{t=1} = e^{2 \pi i q k} = \dfrac 1 {s^q |_{s=1}}$ for $k \in \Z$. The principal branch means to choose $k=0$. --Usagiop (talk) 19:56, 2 October 2022 (UTC)

I found Definition:Power (Algebra)/Complex Number/Principal Branch/Positive Real Base but the contents seem nonsense. --Usagiop (talk) 20:11, 2 October 2022 (UTC)

Well of course everything's nonsense to you isn't it. Perhaps you might like to define it properly then, and cite the source work you got the definition from. --prime mover (talk) 21:59, 2 October 2022 (UTC)

I found a rigorous page Definition:Principal Branch of Complex Number. I believe it is now OK. --Usagiop (talk) 22:20, 2 October 2022 (UTC)