Vito Prosciutto: Teaching community college math on the road to a PhD.

Tuesday, May 24, 2005

(Basic) Complex Analysis: Do I have this right? 

Working from Cain Complex Analysis, Exercise 2.10:
At what points is the function f given by f(z)=x3+i(1-y)3 analytic? Explain.
Since the points where the function has a derivative are two intersecting lines, my assumption is that there is no point where the function is analytic. Am I correct on this or am I making some really basic error?

Update: Hmm, going back over the function, the derivative that Cain gives for (1-y)3 is wrong. I tried it using the chain rule, by evaluating (1-y)3 and doing the derivative of the resulting polynomial and I even went to Quick Math and let their solver do it. And I was right, Cain was wrong: The derivative is -3(1-y)2, and not positive three. That and a quick internet search on analytic complex function reveals a definition which doesn't seem to match his. I'm thinking that perhaps I want to work with a different complex analysis book over the summer.

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