I don't understand this estimation lemma example

[blahblah8724]

I don't understand this estimation lemma example :(

We are given the 'curve',

And part of the example is showing that the contour integral over the top semicircle C_R tends to zero.

Apparently we use the estimation lemma and the fact that, |z^2 +1| \geq |z|^2 - 1 to show,

\left| \int_{C_R} \frac{e^{iz}}{z^2 + 1} dz \right| \leq \int_0^\pi \frac{e^{-Rsin(t)}}{R^2 - 1} dt \leq \frac{2\pi R}{R^2 - 1} \to 0 as R \to \infty

However I don't understand the part where e^{iz} 'goes to' e^{-Rsin(t)}, is this some sort of parameterisation on the curve C_R?

Help would be much appreciated!

Thanks

 

[micromass]

Note that

|e^{a+bi}|=|e^a||e^{bi}|=e^a

Now write out

e^{iz}=e^{iR(\cos(t)+i\sin(t))}

 

[blahblah8724]

Note that

|e^{a+bi}|=|e^a||e^{bi}|=e^a

Now write out

e^{iz}=e^{iR(\cos(t)+i\sin(t))}

Ah thank you that makes sense, but then for the next '\leq' step surely as the arclength of C_R is \pi R and as e^{-Rsin(t)} is always less than 1 for t \in [0,\pi], then by the estimation lemma it should be \frac{\pi R}{R^2 - 1} instead of \frac{2\pi R}{R^2 - 1}?