Math 519 (Spring 2012) Assignment 1
Due Wednesday, February 1
1. Prove that |z|2 = z z̄ for z ∈ C.
2. Use the CREs to show that f (z) = e−y sin x − ie−y cos x is entire.
3. Where are the Cauchy-Riemann equations satisfied for g(z) = zIm(z)?
4. S&S (Stein & Shakarchi) 1.1.
5. S&S 1.3. These are sometimes called the nth cyclic roots of ω.
6. S&S 1.10.
7. S&S 1.11. It may be useful to notice that ∆f = ∆u + i∆v.
8. S&S 1.13ab (not c).
9. S&S 1.24.
Also, read (but don’t do!) 1.9 and 1.18.
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