The definition for the complex power [math]z[/math] of a complex number [math]w[/math] is given below.[br]This definition uses the multi-valued complex natural logarithm [math]log\left(\right)[/math] which is itself defined by:[br][math]log\left(z\right)=Log\left(z\right)+2\pi ik[/math] for [math]k\in\mathbb{Z}[/math][br]where [math]Log\left(\right)[/math] is our principal branch defined by:[br][math]Log\left(z\right)=ln\left(|z|\right)+Arg\left(z\right)[/math] were [math]Arg\left(\right)[/math] is our principal branch of the argument function and [math]ln\left(\right)[/math] is the real nature logarithm.[br] [br]The complex power is defined by:[br][math]w^z=exp\left(z\cdot log\left(w\right)\right)[/math][br]where[br][math]exp\left(Z\right)=\sum^{\infty}_{i=0}\frac{Z^i}{i!}[/math][br]which gives[br][math]w^z=exp\left(z\cdot\left(Log\left(w\right)+2\pi ik\right)\right)[/math] for [math]k\in\mathbb{Z}[/math][br][br]The red points give are for k positive and the green points are for k negative.[br][br]Drag [math]w[/math] and [math]z[/math] about to see how all the values of [math]w^z[/math] changes.