In general [math]e^z[/math] has multiple values for a given complex z.[br][br]The multiple values are found as follows:[br][math]e^z=exp\left(z\left(1+2\pi ik\right)\right)[/math] for [math]k\in\mathbb{Z}[/math][br][br]This comes from a definition for the complex power [math]z[/math] of a complex number [math]w[/math] given below. 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 s slider moves the complex point [math]z[/math] around the circle radius [math]r[/math] center the origin. Our principal value of [math]e^z[/math] is [math]exp\left(z\right)[/math] and the other values are show in red and green.[br][br]The red point are for positive k and the green for negative k.