Real and Complex Analysis (Rudin)

APPROXIMATIONS BY RATIONAL FUNCTIONS 277
10 Suppose Q is a region, I E H(Q), and I t= O. Prove that I has a holomorphic logarithm in Q if and
only if/has holomorphic nth roots in Q for every positive integer n.
11 Suppose that I. E H(Q) (n = 1, 2, 3, ... ),/is a complex function in Q, and/(z) = Iim._", I.(z) for
every z E Q. Prove that Q has a dense open subset V on which I is holomorphic. Hint: Put
cp = sup 1/.1. Use Baire's theorem to prove that every disc in Q contains a disc on which cp is
bounded. Apply Exercise 5, Chap. 10. (In general, V ~ Q. Compare Exercises 3 and 4.)
12 Suppose, however, that/is any complex-valued measurable function defined in the complex plane,
and prove that there is a sequence of holomorphic polynomials p. such that Iim._", p.(z) = I(z) for
almost every z (with respect to two-dimensional Lebesgue measure).