Jump problem for holomorphic functions

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Problem 2.16.9 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.16 "Analytic Continuation"

Check your answers in Exercise 8 against the special case

E = [-1,1]

Problem 18.10 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 18 "Computing Values of Regular Branches of Multivalued Functions. Laurent Series for Regular Branches."

Let $P(z)$ be a polynomial. Prove that any function $f(z)$ , regular in the entire complex plane with a cut along the positive part of the real axis, continuous up to the boundary of this domain, and satisfying, for all real $x > 0$ , the condition

f(x+i 0)-f(x-i 0) = P(x),

has the form

f(z) = -\frac{P(z)}{2 \pi i} h(z)+g(z),

where $g(z)$ is a function regular in the entire complex plane, and $h(z)$ is a regular branch of the multivalued function $\operatorname {Ln} z$ in the plane with a cut along the positive part of the real axis.

Problem 18.11 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 18 "Computing Values of Regular Branches of Multivalued Functions. Laurent Series for Regular Branches."

Find the general form of a function $f(z)$ , regular in the entire complex plane with a cut along the positive part of the real axis, continuous up to the boundary of this domain except at the point $z = 0$ , and satisfying one of the conditions $(x > 0)$ :

(1)

$f(x+i 0)-f(x-i 0) = (1+x)^{2} \ln x$ ;

(2)

$f(x+i 0)-f(x-i 0) = \ln^{2} x$ ;

(3)

$f(x+i 0)-f(x-i 0) = \sin \sqrt{x}$ ;

(4)

$f(x+i 0)-f(x-i 0) = \frac{\sin (\alpha \ln x)}{x-1}$ ;

(5)

$f(x+i 0)-f(x-i 0) = \frac{x+1}{\ln^{2} x+\pi^{2}}$ ;

(6)

$f(x+i 0)-f(x-i 0) = \sqrt{x} \ln x$ ;

(7)

$f(x+i 0)-f(x-i 0) = \frac{\ln x}{\sqrt[3]{x}}$ ;

(8)

$f(x+i 0)-f(x-i 0) = \frac{\sqrt{x} \ln x}{\ln^{2} x+\pi^{2}}(x+1)$ .

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.