Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to the primary course text by Brown and Churchill. Unless otherwise indicated, all parts of a problem (a), (b), etc. should be completed. You are encouraged to work on these problems with other classmates, although the solutions you turn in should be YOUR OWN WORK.
Important: Please list the names of any students or faculty who you worked with on the assignment. Also cite any references (website, other texts, etc.) that you may have used while working on this assignment.
Sections 19 and 20 (p. 62 - 63): #1b, 1c, 3, 9
Note: To paraphrase the instructions for #9, set up the limit definition for the derivative of the given function at z0 = 0. Then approach the origin in three different ways. Along the real and imaginary axes you should get 1, while approaching on the diagonal Re(h) = Im(h) you should get -1. Conclude that f'(0) does not exist.
Sections 21 - 23 (p. 71 - 73): #1a, 1d, 3b, 3c, 4a, 4b, 6, 7, 8
Hints: For #4, use Examples 1 and 2 on p. 70 as a guide. For #6 you will need to use the limit definitions of the partial derivatives to calculate ux, uy, vx, and vy at the origin. The point of this problem is to provide an example where the Cauchy-Riemann equations are satisfied yet the derivative fails to exist. The SCD Theorem does not apply because the third condition is not satisfied. Bonus points if you can rigorously explain why.