Identity in proof

Question

I'm reading a proof and at some point author makes the following claim: $e^{2\sqrt{a}k}=1 \iff 2\sqrt{a}=2k\pi i$.

It was used to solve a differential equation. I'm don't see why it is true, it doessn't seem to be trivial at all. It looks like it would follow from a trigonometric identity.

Answer 1

Do you know the relation $e^{ix}=\cos(x)+i\sin(x)$?

We want $\sin(x)=0$ and $\cos(x) =1$. Which is true iff $x=2k\pi \, (k\in\mathbb{Z})$.

Are you sure you are not missing a $k$ on your left side of your equation?

Answer 2

The above claim is not o.k. !

For $z \in \mathbb C$ we have

$e^z=1$ iff there is $j \in \mathbb Z$ such that $z=2 j \pi i$

Thus

$e^{2\sqrt{a}k}=1$ iff there is $j \in \mathbb Z$ such that $2\sqrt{a}k=2 j \pi i$