The Complex Exponential: the Functional Equation, Conjugates, Real Arguments and the Modulus

In the setting of Complex Analysis: Standing Notation, let $\exp:\mathbb{C}\to\mathbb{C}$ be the complex exponential function, and let $z,w\in\mathbb{C}$.

(i) Zero. $\exp 0=1$.

(ii) The functional equation.

$$\exp(z+w)=\exp z\cdot\exp w .$$

(iii) Reciprocals. $\exp z\neq0$, and

$$\exp(-z)=\frac{1}{\exp z}.$$

(iv) Conjugates.

$$\overline{\exp z}=\exp\overline{z}.$$

(v) Real arguments. For every real number $x$,

$$\exp x=\exp_{\mathbb{R}}(x);$$

in particular $\exp x$ is a real number.

(vi) The modulus.

$$|\exp z|=\exp_{\mathbb{R}}(\operatorname{Re}z),$$

and $|\exp(iy)|=1$ for every real number $y$.