Here is my question: Let be analytic and suppose that . If , find .
My proof so far follows:
By differentiating with respect to x and y, we see that , , , and . In order to move back to v, I will integrate on and as follows:
To solve for c, and , so
and now this is the really dumb part... I am not sure if I am supposed to substitute this literally as u + i*v, using the u as given and the v that I have derived, substituting (0,1) for i, or do something else entirely. I feel that I am nearly at the solution, but I am losing it here in something that should be quite simple. I will appreciate your help. Thanks.
But don't I have to put the v in there somewhere, too, since that's what I've had to find? And I'm trying to solve for c? Since f=u+iv to start with, don't they go together? That's where I am getting confused. I have discussed this problem with classmates and nobody mentioned coming up with a contradiction. I know that we are supposed to solve for c and I really thought that the answer was either 2 or 2i, but for the life of me, I can't get there.