## Complex Exponentials (3)

(Note: this is a quick and dirty introduction to exponentials. I will not bother with any epsilons or deltas, but for those of you who are concerned about such things, be assured that all convergence is uniform, all sequences are Cauchy, etc. The series that defines the exponential is about as well behaved as an infinite series can be.)

The function \[ e^{x} \]is defined to be the limit as \( n \) approaches infinity of \[ ( 1 + \frac{x}{n})^{n}: \]

exp(x) = \[ e^{x} = \lim_{n \to \infty} (1 + \frac{x}{n})^{n} .\]

Consider first the exponential of a constant, a:

\[ e^{a} = \lim_{n \to \infty} (1 + \frac{a}{n})^{n} = \sum_{k=0}^{\infty} \frac{a^{k}}{k!}. \]

If b is another constant, what is \[ e^{a}e^{b} ? \]

Let's go back to the definition of exponential:

\[ e^{a}e^{b} \ = \lim_{n \to \infty} f(n) .\]

Where:

\[ f(n) = (1 + \frac{a}{n})^{n} (1 + \frac{b}{n})^{n} = ( 1 + \frac{ a + b}{n} + \frac{ab}{n^{2}})^{n} \]

\[ = \sum_{k=0}^{n} \binom{n}{k} c^{k}, \]

Consider the exponential of a purely imaginary number:

\[ z = i\alpha \]

with \[ \alpha \in \mathbb{R}\ .\]

\[ e^{i\alpha} = \lim_{n \to \infty } (1 + \frac{i\alpha}{n})^{n} \]

\[ = \sum_{k=0}^{\infty }\frac{i\alpha^{k}}{k!} . \]