Can someone explain this equality to me (complex variables)

[richyw]

Homework Statement

I hate to upload the whole problem, but I am trying to evaluate an indefinite integral, and I can follow the solution until right near the end. The example says that for a point on C_R|e^{-3z}|=e^{-3y}\leq 1. I don't understand how they can say this. Below is the question, with a drawing of the region. I have highlighted the step that I do not understand.

http://media.newschoolers.com/uploads/images/17/00/70/52/45/705245.png

Homework Equations

The Attempt at a Solution

I might be missing something easy, but I can't see how this is true!

 

[LCKurtz]

$e^{3iz}=e^{3i(x+iy)}=e^{-3y+3ix}=e^{-3y}e^{3ix}$ and $|e^{i3x}|=1$.

 

[az_lender]

exp(3iz) = exp(3ix - 3y) = exp(-3y) exp(3ix)
The magnitude of this complex number is |exp(-3y)| times 1, because exp(3ix) = cos(3x) + i sin(3x), and |exp(3ix)| is the sum of a squared cosine and a squared sine of the same argument. And then of course |exp(-3y)| = exp(-3y) because e-to-the-anything is always positive.

Why is exp(-3y) <= 1? Because exp(0) = 1, and on the given semicircular path, y is non-negative.