(adsbygoogle = window.adsbygoogle || []).push({}); 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 [itex]C_R[/itex] tends to zero.

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

[itex]\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[/itex] as [itex] R \to \infty [/itex]

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

Help would be much appreciated!

Thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I don't understand this estimation lemma example

**Physics Forums | Science Articles, Homework Help, Discussion**