
1.7: The Exponential Function


We have Euler's formula: $$e^{i\theta} = \cos (\theta) + i \sin (\theta)$$. We can extend this to the complex exponential function $$e^z$$.

Definition

For $$z = x + iy$$ the complex exponential function is defined as

$$e^z = e^{x + iy} = e^x e^{iy} = e^x (\cos (y) + i \sin (y))$$.

In this definition $$e^x$$ is the usual exponential function for a real variable $$x$$.

It is easy to see that all the usual rules of exponents hold:

1. $$e^0 = 1$$
2. $$e^{z_1 + z_2} = e^{z_1} e^{z_2}$$
3. $$(e^z)^n = e^{nz}$$ for positive integers $$n$$.
4. $$(e^z)^{-1} = e^{-z}$$
5. $$e^z \ne 0$$
It will turn out that the property $$\dfrac{de^z}{dz} = e^z$$ also holds, but we can’t prove this yet because we haven’t defined what we mean by the complex derivative $$\dfrac{d}{dz}$$.
Here are some more simple, but extremely important properties of $$e^z$$. You should become fluent in their use and know how to prove them.
6. $$|e^{i \theta}| = 1$$
Proof.
$$|e^{i \theta}| = |\cos (\theta) + i \sin (\theta)| = \sqrt{\cos ^2 (\theta) + \sin ^2 (\theta)} = 1$$
7. $$|e^{x + iy}| = e^x$$ (as usual $$z = x + iy$$ and $$x, y$$ are real).
Proof. You should be able to supply this. If not: ask a teacher or TA.
8. The path $$e^{it}$$ for $$0 < t < \infty$$ wraps counterclockwise around the unit circle. It does so infinitely many times. This is illustrated in the following picture.

The map $$t \to e^{it}$$ wraps the real axis around the unit circle.