The Complex Exponential Function

The Complex Exponential Function

In calculus we are first introduced to the real exponential function $f(x) = e^x$ which is defined for all $x \in \mathbb{R}$ and whose range is $(0, \infty)$. The graph of $f$ is given below.

We would now like to extend this function to allow inputs of all complex numbers (and not just real numbers), i.e., we would like to define the complex exponential function $f(z) = e^z$ for all $z \in \mathbb{C}$. For an idea of how to do this, first recall the following Maclaurin series representation for the functions $\cos x$ and $\sin x$:

(1)
\begin{align} \quad \cos x = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ... \end{align}
(2)
\begin{align} \quad \sin x = \sum_{n=0}^{\infty} \frac{(-1)^nx^{2n + 1}}{(2n + 1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... \end{align}

These series converge to $\cos x$ and $\sin x$ respectively for all $x \in \mathbb{R}$. Now, the Maclaurin series representation for the real-valued function $e^x$ is:

(3)
\begin{align} \quad e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ... \end{align}

This series also converges to $e^x$ for all $x \in \mathbb{R}$. Now let $z = x + yi$. Then a reasonable definition for allowing imaginary number inputs $yi$ would be:

(4)
\begin{align} \quad e^{yi} = \sum_{n=0}^{\infty} \frac{(yi)^n}{n!} &= 1 + yi + \frac{(yi)^2}{2!} + \frac{(yi)^3}{3!} + \frac{(yi)^4}{4!} + ... \\ & = 1 + yi - \frac{y^2}{2!} - \frac{y^3}{3!} i + \frac{y^4}{4!} + \frac{y^5}{5!}i + ... \\ &= \left (1 - \frac{y^2}{2!} + \frac{y^4}{4!} - \frac{y^6}{6!} + ... \right ) + \left ( y - \frac{y^3}{3!} + \frac{y^5}{5!} - \frac{y^7}{7!} + ... \right ) i \\ &= \cos y + i \sin y \end{align}

Since $y \in \mathbb{R}$, $e^{yi}$ is well defined for each imaginary number $yi$. We will take this to be the definition of $e^{yi}$ even though we have not yet shown the convergence of the complex Maclaurin series above. Now, for a more general complex number $z = x + yi$ we would like for the usual rules of exponents to hold, i.e., we would like $e^{z} = e^{x + yi} = e^x \cdot e^{yi}$ for all $z \in \mathbb{C}$. We formally define this rule below.

 Definition: The Complex Exponential Function is defined for all $z = x + yi \in \mathbb{C}$ by $e^z = e^{x + yi} = e^x \cdot e^{yi} = e^x (\cos y + i \sin y)$.

The values of $e^z$ have a nice geometric interpretation. If $z = x + yi$ then the modulus of the complex number $f(z) = e^z$ is simply $e^x$, and the argument of $f(z) = e^z$ is $y$. Notice that if $z \in \mathbb{R}$ then $y = 0$ and $e^z = e^{x + 0i} = e^x$ which is the real-valued exponential function.

Polar Representations of Complex Numbers and the Exponential Function

Recall that if $z \in \mathbb{C}$ is a nonzero complex number then $z$ has a polar representation $z = r(\cos \theta + i \sin \theta)$ where $\theta = \mathrm{arg} (z)$ is the angle made between $z$ and the positive real axis and $r = \mid z \mid$.

Since $e^{\arg (z) i} = e^{i \theta} = (\cos \theta + i \sin \theta)$ we see that:

(5)
\begin{align} \quad z = r(\cos \theta + i \sin \theta) = re^{i\theta} \end{align}
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