The Complex Exponential Function#

Theorem: Complex Exponential Function

\[ \sum_{n = 0}^{\infty} \frac{z^n}{n!} = 1 + \frac{z^1}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \frac{z^4}{4!} + \cdots \]

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

Proof

We use the ratio convergence test. Let $a_n = \frac{z^n}{n!}$. We have:

\[ \left\vert\frac{a_{n+1}}{a_n}\right\vert = \frac{|z|^{n+1}}{(n+1)!}\frac{n!}{|z|^n} = \frac{|z|}{n+1} \]

Since

\[ \lim_{n \to \infty} \frac{|z|}{n+1} = 0 \lt 1 \]

for all $z \in \mathbb{C}$, we know that $\sum_{n = 0}^{\infty} \frac{z^n}{n!}$ is absolutely convergent for all $z \in \mathbb{C}$.

Definition: Complex Exponential

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

\[ \exp (z) \overset{\text{def}}{=} \sum_{n = 0}^{\infty} \frac{z^n}{n!} \]

Notation

\[ \exp z \qquad \exp(z) \qquad \mathrm{e}^z \]

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 $2\pi \mathrm{i}$:

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

Proof

TODO

Theorem: Euler's Formula

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

\[ \mathrm{e}^{\mathrm{i}\varphi} = \cos \varphi + \mathrm{i} \sin \varphi \]

Proof

TODO

Theorem: Exponent Arithmetic

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

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

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:

\[ |\mathrm{e}^z| = \mathrm{e}^{\operatorname{Re}(z)} \]

Proof

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

$$
\begin{aligned}

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

\end{aligned}
$$

Theorem: Conjugation of the Complex Exponential

The conjugate of the complex exponential is the complex exponential of the conjugate:

\[ \overline{\mathrm{e}^z} = \mathrm{e}^{\overline{z}} \]

Proof

PROOF

Theorem: Complex Exponential via Complex Limit

The complex exponential function $\mathrm{e}^z$ is equal to the following limit:

\[ \mathrm{e}^z = \lim_{n \to \infty}\left(1 + \frac{z}{n}\right)^n \]

Proof

We need to prove that for each $\varepsilon \gt 0$, there exists some $N \in \mathbb{N}_0$ such that

\[ \left\vert \left(1 + \frac{z}{n}\right)^n - \sum_{k=0}^{\infty}\frac{z^k}{k!} \right\vert \lt \varepsilon \]

for all $n \ge N$.

TODO