It turns out that there was yet another issue with question 3 (see below), so I have decided that I will move it to next week's assignment. Also, I have the feeling that there are a lot of you who could use more time with the other two questions, so I will postpone the deadline to Monday at 11.59 PM EDT. (I will make both of these changes in Crowdmark later on tonight.) You are of course encouraged to finish earlier if you can as the next assignment will be due on Friday (and will be posted later on tonight or in the wee hours of the morning tomorrow).

Question 3 needs to be modified as follows: First, the function f is required to be nonzero; second, the curve C_R actually should be the full circle, not just a segment as currently written. With this, there is a hint to use the proof of Liouville's Theorem, not just the statement itself.

A couple more hints: Question 1: when I say 'evaluate the integral of e^(iz^2)/(1 + z^8)', this doesn't mean that you can actually evaluate that integral. Rather, this hint is more like a hint in a treasure hunt that says, Go down that road -- it doesn't mean you will find the treasure at the end of that road, but rather that at some point you will see something else which will allow you to find the treasure. In other words, if you push on the integral of e^(iz^2)/(1 + z^8), you should see after a while how to obtain the original integral. (It requires closing with a wedge contour, and there is really only one which is reasonable -- you want the integral over the additional line segment to be very close to the original integral; it won't be quite identical though!)

Question 2: you need to use an indented contour, but a keyhole contour won't quite work. Try an indented 'wedge' (though it's wide enough it might not exactly look like a 'wedge'...).

I hope that helps -- please let me know if you have any further questions!

Nathan