Find a conformal mapping of the left half-plane and choose

Question

Question 1

(1) Determine whether F (given below) is conservative in the given region D. D is defined to bethe entire plane.

(2) If the vector field is conservative, find a potential

(3) Verify the potential.

F = y³ i + (3xy² - 4)j

Question 2

Evaluate the integral of the following function over the closed path. Consider the curve to be oriented counterclockwise.
f(z) = 2z/((z-i)); γ is the circle |z-i|=3

(1) Is Cauchy's theorem applicable?
(2) Determine the value of the integral.

Question 3

(1) Find a conformal mapping of the left half-plane onto the disk |w| < 4 in the w-plane.

(2) Choose a point to verify that the left half-plane maps to the interior of the circle.