Illinois Institute of Technology Department of Applied Mathematics Math 333: Matrix Algebra and Complex Variables Homework 5 Due Date: Wednesday, March 8, 2017 Note: show full steps to get full credit. 1. Consider function f (z) = z 3 . (a) Use the definition of derivatives to calculate the derivative of f , the result should be a function of z. (b) Use Cauchy-Riemann equation to show that f is an entire function. (c) Find the derivative of f (z) in terms of u and v. 2. At what points are the following functions not analytic (a) z z3i . (b) z 2 2iz . z 2 +4 3. Compute f 0 if f = 4z 3 5z+1 2z1 . 4. Show that the following functions are not analytic at any point. (a) f (z) = y + ix (b) f (z) = z2 (c) f (z) = 2x2 + y + i(y 2 x) 5. Show that (ez )0 = ez . 6. Express f (z) = e2z in u + iv form. 7. Solve ez1 = ie2 . 8. Express sin(2i) in a + ib form. 9. Find all values of z such that cos z = 3i. 1