The complex exponential function $\exp:\,\mathbb{C}\to \mathbb{C}$ may be defined in many equivalent ways: Let $z = x\!+\!iy$ where $x,\,y\in\mathbb{R}$ .

• $\displaystyle\exp{z} \;:=\; e^x(\cos{y}+i\sin{y})$
• $\displaystyle\exp{z} \;:=\; \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n$
• $\displaystyle\exp{z} \;:=\; \sum_{n = 0}^\infty\frac{z^n}{n!}$

The function gets all complex values except 0 and is periodic having the prime period (the period with least non-zero modulus) $2\pi i$ . The $\exp$ is holomorphic, its derivative $$\frac{d}{dz}e^z \;=\; e^z,$$ which is obtained from the series form via termwise differentiation, is similar as in $\mathbb{R}$ .

So we have a fourth way to define

• $\exp{z} \;:=\; w(z)$

Some formulae: $$|e^z| \;=\; e^x, \quad \arg{e^z} \;=\; y+2n\pi\quad(n = 0,\,\pm1,\,\pm2,\,\ldots),$$ $$\mbox{Re}(e^z) \;=\; e^x\cos{y}, \quad \mbox{Im}(e^z) \;=\; e^x\sin{y}$$