Consistency and well-definedness of analytic continuation

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

(a)(b)

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

(a)

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

(b)

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

Problem 20.1 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

Prove that a function regular in a domain $G$ is a function analytic in the domain $G$ .

Problem 20.2 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

Prove that if a function analytic in a domain $G$ does not depend on the shape of the piecewise-smooth oriented curve along which the analytic continuation is carried out, but depends only on its endpoint, then this function is regular in the domain $G$ .

Problem 20.15 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

Prove that if the initial elements of two functions analytic in a domain $G$ are equivalent, then these functions are identically equal.

Problem 20.16 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

Prove that if some element of a function analytic in a domain $G$ is equivalent to zero, then this function is identically equal to zero.

Problem 20.23 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

Let the function $f$ be regular in a neighborhood of the point $z = 0$ , and let the sequence $\left\{ f^{(n)}(z)\right\}$ converge there uniformly, and

\lim _{n \rightarrow \infty } f^{(n)}(0) = 1

Prove that the limit function of this sequence can be analytically continued into $\mathbb {C}$ .

References

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.

[ШПК15] Шабунин М.И., Половинкин Е.С., Карлов М.И. Сборник задач по теории функций комплексного переменного. 4-е изд., БИНОМ. Лаборатория знаний, 2015.