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Details 5

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5.5 [Use Morera's Theorem. You may use without proof that f is continuous. ]

5.9

5.13 [Adapt the proof of Liouville's Theorem.]

5.15

5.17

5.5. Define f : D[0, 1] ® C through

f (z) := ò_{[0, 1]} dw/1 - wz  

(the integration path is from 0 to 1 along the real line). Prove that f is holomorphic in the unit disk D[0, 1].

5.9. Find a region on which f (z) = exp(1/z) has an antiderivative. (Your region should be as large as you can make it. How does this compare with the real function f(x) = e^1/x?)

5.13. Suppose f is entire and |f (z)| £ Ö|z| for all z Î C. Prove that f is identically 0. (Hint Show first that f is constant.)

5.15. Suppose f is entire with bounded real part, i.e., writing f (z) = u(z) + i v(z), there exists M > 0 such that |u(z)| £ M for all z Î C. Prove that f is constant. (Hint: Consider the function exp(f (z)).)

5.17. Suppose f : D[0,1] ® D[0,1] is holomorphic. Prove that for |z| < 1,

|f’ (z)| £ 1/1 - |z|.