Harmonic functions of a prescribed form

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Problem 5.9.4 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.9 "Harmonic Functions"

Let $s$ be a real number, and let the function $u$ be defined in $\mathbb {C} \setminus (-\infty , 0]$ by

u(re^{i\theta }) = r^{s} \cos s\theta \qquad (r > 0, \ -\pi < \theta < \pi ) .

Prove that $u$ is a harmonic function.

Problem 5.1.17 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 5.1 "Analytic and Harmonic Functions"

Find the most general harmonic function of the form $f(\left|z\right|)$ , $z \in \mathbb {C} \setminus 0$ . Which of these $f(\left|z\right|)$ have a single valued harmonic conjugate?

Problem 5379 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.1 "The Cauchy--Riemann Equations"

Prove that the equation $u(x,y) = C$ can represent a family of lines of equal potential only when the ratio of the quantities

\frac{\partial ^{2} u}{\partial x^{2}}+\frac{\partial ^{2} u}{\partial y^{2}} \quad \text{and} \quad \left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial u}{\partial y}\right)^{2}

is a function of $u$ alone.

Problem 5380 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.1 "The Cauchy--Riemann Equations"

Prove that the curves $x^{n}+y^{n} = C$ can be lines of equal potential only for $n = 1$ and for $n = 2$ .

Problem 3.65 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Some Particular Elementary Mappings"

Along which curves of the $z$ -plane is the real part of $z^{2}$ constant? Along which curves is the imaginary part constant? The two families of curves form an orthogonal system; why?

Problem 5.14 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 5 "Differentiability of Functions. Harmonic Functions"

(1)(2)(3)(4)

Find all harmonic functions of the form

(1)

$u = \varphi \left(x^{2}+y^{2}\right)$ ;

(2)

$u = \varphi \left(x^{2}-y^{2}\right)$ ;

(3)

$u = \varphi \left(\frac{y}{x}\right)$ ;

(4)

$u = \varphi \left(\frac{x^{2}+y^{2}}{x}\right)$ .

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.

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.

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