# $e^z, \ \ z \in \mathbb{C}$ isn't invertible

Could you tell me why the function $\mathbb{R}^2 \ni z \rightarrow e^z \in \mathbb{R}^2$, complex exponential, is not invertible globally. On a horizontal strip $[\ i y, \ i(y+2 \pi))$ its inverse is $\ln z$.

I found this example on this forum, but there is no explanation if this fact. Or at least it is not explanatory enough for me to understand.

I would really appreciate all your insight.

Thank you.

• Try to define a continuous inverse as you travel around the unit circle, and you may see where the problem arises. – Gerry Myerson Nov 11 '13 at 9:08
• Do you know, that $e^{ix} = \cos(x) + i\sin(x)$ for $x\in\mathbb{R}$? – roman Nov 11 '13 at 9:11
• A function must be injective to be invertible. – anon Nov 11 '13 at 9:27

I think you have to understand that a general function $f : A\to B$ to be invertible need first to be injective. Why that? Well, because recall for each $x\in A$ there must be unique $y\in B$ with $y=f(x)$. If $f$ is not injective, then there are $x_1,x_2\in A$, $x_1\neq x_2$ with $f(x_1)=f(x_2)$. Now, try defining the inverse, you can easily see that if $g : B \to A$ must be such that $g(f(x))=x$, then you have a problem, because you would have $g(f(x_1))=x_1$ and at the same time $g(f(x_2))=x_2$, but $f(x_1)=f(x_2)=y$, so for this $y$ there are two values in $A$, so $g$ isn't a function.
If a function $f: \Bbb C\to \Bbb C$ is periodic, that is there is $k\in \Bbb C$ with $f(z+k)=f(z)$, then it's clearly not injective, hence not invertible. As you can see
$$\exp(z+2\pi i)=\exp(x+iy+2\pi i)=e^x(\cos (y+2\pi) + i\sin(\theta+2\pi))=\exp(z),$$
This means that since the function repeats when you add $2\pi i$ to the point, in every horizontal strip of vertical length less than $2\pi$ the function doesn't have duplicate values: it's injective, hence invertible. More precisely, let $\alpha \in \mathbb{R}$, then you let $S_{\alpha}=\{z \in \mathbb{C} : \alpha < \Im(z) < \alpha + 2\pi\}$, in this region $\exp$ is injective and invertible.
Periodic functions are not invertible, and $e^z$ is $2\pi i$-periodic.