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 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)