# Quaternion algebra using analytic continuation

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.

• 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 $f(a+b\mathbf{u})=c+d\mathbf{u}$. – arctic tern Mar 4 at 4:30
• You can see: math.stackexchange.com/questions/2095894/… – Emilio Novati Mar 4 at 10:48
• @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? – Benjamin Thoburn Mar 4 at 18:49