Next I need to estimate the integral over the semi circle gamma R of my function f.

Remember gamma R was simply the semi circle of radius R centered at the origin.

Plus we're only really interested for what happens when R goes to infinity.

What we want to show is that the integral actually goes to 0 as R goes to infinity,

so it doesn't really matter then.

In order to show that some complicated expression goes to 0 as R

goes to infinity, really suffices to show that the absolute value

of that complicated expression is bounded above by some constant for

which it is easier for us to show that it goes to 0.

So as long as this constant goes to 0, this integral,

which is stuck between 0, and this constant that goes to 0 then is

sandwiched in between 0 and the constant and therefore has to go to 0 as well.

So we're going to try to find an upward bound for

the absolute value of the integral and show that upper bound goes to 0, and

then that integral is first to go to 0 as well.

We need to show that the absolute value of the integral is bounded above by

a constant wherer the constant goes to 0.

How do we estimate the absolute value of an integral?

Remember what we learned a while ago.

The absolute value of a contour integral is bounded above

by the length of the curve over which we're integrating times the maximum of

the absolute value of f, for z's that are on the curve.

So let's use that.