Extendability of holomorphic functions past their domain

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Problem 5.6.2 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.6 "Analytic and Meromorphic Functions"

True or false: A function f(z)f(z) analytic on ∣z−a∣<r\left|z-a\right|<r and continuous on ∣z−a∣≤r\left|z-a\right| \leq r extends, for some δ>0\delta >0, to a function analytic on ∣z−a∣<r+δ\left|z-a\right|<r+\delta? Give a proof or a counterexample.

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Problem 5.5.16 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 5.5 "Series and Normal Families"

For ∣an∣<1\left|a_n\right|<1,

f(z)=∏n=1∞an−z1−aˉnz. f(z) = \prod _{n=1}^{\infty } \frac{a_n-z}{1-\bar a_n z}.
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(a)

Show that ff defines a holomorphic function on D={z:∣z∣<1}D = \left\{ z : \left|z\right|<1\right\}.

(b)

Prove that ff does not have an analytic continuation to any larger disk {z:∣z∣<r}\left\{ z:\left|z\right|<r\right\} where r>1r>1.

Problem 2.19.7 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.19 "Holomorphic Fourier Transforms"

Let ff be a complex function on a closed set EE in the plane. Prove that the following two conditions on ff are equivalent:

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(a)

There is an open set Ω⊃E\Omega \supset E and a function F∈H(Ω)F \in H(\Omega ) such that F(z)=f(z)F(z) = f(z) for z∈Ez \in E.

(b)

To each α∈E\alpha \in E there corresponds a neighborhood VαV_{\alpha } of α\alpha and a function Fα∈H(Vα)F_{\alpha } \in H\left(V_{\alpha }\right) such that Fα(z)=f(z)F_{\alpha }(z) = f(z) in Vα∩EV_{\alpha } \cap E. (A special case of this was proved in Theorem 19.9).

References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.
[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.
[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.