clockwise and counterclockwise integration
This is one of the problems that I have to do for Monday, and I really don't get it. I didn't understand the explainations in class. If someone could help me with these, I would really appreciate it. I think something like this could be on the final as well, so if you could really explain how to do every step that would be fantastic!
1. Furnishing a sketch of the curve you are integrating over, integrate the following:
a. (cos 4z)/(z^3(4z - (pi)) counterclockwise around the circle |z - 1| = 1/2
b. Integral (with c at the bottom) of 1/ (z^2 - 1) dz where c is the curve with the given orientations. HINT: C = C1 union C2 where C1: |z - 1| = 1, counterclockwise, C2: |z + 1| = 1 clockwise.
c. e^z / (ze^z - 2iz) where C is the circle |z| = 0.5, counterclockwise.
d. 4(z + 2i)^(-1) + 2(z + 4i)^(-1) clockwise around the circle |z - 1| = 2.5
e. (z^3)(e^z) / (2z - 1)^3 counterclockwise around the unit circle.