As for complex variables, do we use analytic continuation to find things like $sin(j)$, $i^k$, and so on? Is there another method or do these expressions even have values at all.

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    $\begingroup$ Note that the square roots of negative one in the quaternions are precisely the unit vectors $\mathbf{u}$. Therefore, if $f$ is defined by a Laurant series and $f(a+bi)=c+di$, then plugging $a+b\mathbf{u}$ into $f$ (where $\mathbf{u}$ is a unit vector) yields the value $f(a+b\mathbf{u})=c+d\mathbf{u}$. $\endgroup$ – arctic tern Mar 4 at 4:30
  • $\begingroup$ You can see: math.stackexchange.com/questions/2095894/… $\endgroup$ – Emilio Novati Mar 4 at 10:48
  • $\begingroup$ @Emilio thanks, I wasn't even sure that they existed and I guess that partially answers my question. Are there any complexities in typical methods of analytic continuation that come from non commutativity? $\endgroup$ – Benjamin Thoburn Mar 4 at 18:49

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