The Complex Exponential Function

Theorem: Complex Exponential Function

The complex power series

$$\sum _{n=0}^{\infty }\frac{z^n}{n!}=1+\frac{z^1}{1!}+\frac{z^2}{2!}+\frac{z^3}{3!}+\frac{z^4}{4!}+\cdots$$

converges absolutely for all $z\in \mathbb{C}$.

PROOF

TODO

Definition: Complex Exponential

The complex exponential function is the complex function $\exp :\mathbb{C}\to \mathbb{C}$ defined by the aforementioned complex power series:

$$\exp (z)\stackrel{\text{def}}{=}\sum _{n=0}^{\infty }\frac{z^n}{n!}$$

NOTATION

$$\exp z\exp (z){\mathrm{e}}^z$$

Theorem: Complex Exponential Extends Real Exponential

The restriction of the complex exponential function to the real numbers $\mathbb{R}$ is the real exponential function.

PROOF

TODO

Theorem: Periodicity of the Complex Exponential

The complex exponential function is periodic with period $2\pi \mathrm{i}$:

$$\exp (z+2k\pi \mathrm{i})=\exp (z)\forall z\in \mathbb{C},\forall k\in \mathbb{Z}$$

PROOF

TODO

Theorem: Euler's Formula

For every $\phi \in \mathbb{R}$, the real part of the complex exponential function ${\mathrm{e}}^{\mathrm{i}\phi }$ is the real cosine of $\phi$ and the imaginary part is the real sine of $\phi$:

$${\mathrm{e}}^{\mathrm{i}\phi }=\cos \phi +\mathrm{i}\sin \phi$$

PROOF

TODO

Theorem: Exponent Arithmetic

The complex exponential function has the following property for all $z,w\in \mathbb{C}$:

$${\mathrm{e}}^{z+w}={\mathrm{e}}^z\cdot {\mathrm{e}}^w\forall z,w\in \mathbb{C}$$

PROOF

TODO

Theorem: Modulus of the Complex Exponential

For all $z\in \mathbb{C}$, the modulus of the complex exponential function of $z$ is the real exponential of its real part:

$$|{\mathrm{e}}^z|={\mathrm{e}}^{\mathrm{Re}(z)}$$

PROOF

Let $x=\mathrm{Re}(z)$ and $y=\mathrm{Im}(z)$.

$$\begin{array}{rl}|{\mathrm{e}}^z|=\left|{\mathrm{e}}^{x+\mathrm{i}y}\right|& =\left|{\mathrm{e}}^x\cdot {\mathrm{e}}^{\mathrm{i}y}\right|\\ & =\left|{\mathrm{e}}^x\cdot \cos y+\mathrm{i}\sin y\right|\\ & =\left|{\mathrm{e}}^x\sqrt{{\cos }^{2}y+{\sin }^{2}y}\right|\\ & =\left|{\mathrm{e}}^x\cdot \sqrt{1}\right|\\ & =\left|{\mathrm{e}}^{\mathrm{Re}(z)}\right|={\mathrm{e}}^{\mathrm{Re}(z)}\end{array}$$