Harmonicity of expressions built from analytic functions

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Problem 5.1.3 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"

(a)(b)

Let $G$ be a region in $\mathbb {C}$ and suppose $u : G \to \mathbb {R}$ is a harmonic function.

(a)

Show that $\dfrac {\partial u}{\partial x} - i \dfrac {\partial u}{\partial y}$ is an analytic function on $G$ .

(b)

Show that $u$ has a harmonic conjugate on $G$ if and only if $\dfrac {\partial u}{\partial x} - i \dfrac {\partial u}{\partial y}$ has a primitive (anti-derivative) on $G$ .

Problem 5385 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.1 "The Cauchy--Riemann Equations"

Prove the identity:

\frac{\partial ^{2} \left|w(z)\right|}{\partial x^{2}}+\frac{\partial ^{2} \left|w(z)\right|}{\partial y^{2}} = \frac{\left|w'(z)\right|^{2}}{\left|w(z)\right|}.

Problem 3.58 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "The Cauchy-Riemann Differential Equations"

Show that for any analytic function $f(z) = f(x+iy)$

\left(\frac{\partial ^{2}}{\partial x^{2}} + \frac{\partial ^{2}}{\partial y^{2}}\right)\left|f(x+iy)\right|^{2} = 4\left|f'(x+iy)\right|^{2}.

Problem 3.59 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "The Cauchy-Riemann Differential Equations"

Show that for any analytic function of $z = x+iy$

\left(\frac{\partial ^{2}}{\partial x^{2}} + \frac{\partial ^{2}}{\partial y^{2}}\right)\log (1+\left|f(x+iy)\right|^{2}) = \frac{4\left|f'(x+iy)\right|^{2}}{(1+\left|f(x+iy)\right|^{2})^{2}}.

Problem 4.171 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Various Propositions"

Find, if possible, $n+1$ analytic functions $f_{1}(z), f_{2}(z), \ldots , f_{n}(z), f(z)$ , which differ from each other not merely by constant factors, are regular in a domain $\mathfrak {D}$ and for which the relation

\left|f_{1}(z)\right| + \left|f_{2}(z)\right| + \cdots + \left|f_{n}(z)\right| = \left|f(z)\right|

holds in $\mathfrak {D}$ .

Problem 4.172 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Various Propositions"

Find, if possible, two non-constant analytic functions $f(z)$ and $g(z)$ , which are regular in a domain $\mathfrak {D}$ and for which the relation

\left|g(z)\right| = \Re f(z)

holds in $\mathfrak {D}$ .

Remark.

See Problem 58 of Part III.

Problem 3.55.2 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Problem 3.55 "[NO TITLE]"

(continued). Prove that

u_{xx} + u_{yy} = v_{xx} + v_{yy} = 0.

Problem 3.55.4 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Problem 3.55 "[NO TITLE]"

(continued). If $\varphi (x,y)$ has continuous derivatives of the second order, also

\varphi _{xx} + \varphi _{yy} = (\varphi _{uu}+\varphi _{vv})\left|\frac{dw}{dz}\right|^{2}.

Problem 3.55.5 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Problem 3.55 "[NO TITLE]"

(continued). Assume that $a, b, c$ and $d$ are real constants,

ad - bc = 1,

and consider

w = \frac{az+b}{cz+d}.

Then

\frac{\varphi _{u}^{2}+\varphi _{v}^{2}}{\varphi _{x}^{2}+\varphi _{y}^{2}} = \frac{\varphi _{uu}+\varphi _{vv}}{\varphi _{xx}+\varphi _{yy}} = \frac{y^{2}}{v^{2}}.

Problem 2.11.1 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.11 "Harmonic Functions"

Suppose $u$ and $v$ are real harmonic functions in a plane region $\Omega$ . Under what conditions is $u v$ harmonic? (Note that the answer depends strongly on the fact that the question is one about real functions). Show that $u^{2}$ cannot be harmonic in $\Omega$ , unless $u$ is constant. For which $f \in H(\Omega )$ is $\left|f\right|^{2}$ harmonic?

Problem 2.11.2 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.11 "Harmonic Functions"

Suppose $f$ is a complex function in a region $\Omega$ , and both $f$ and $f^{2}$ are harmonic in $\Omega$ . Prove that either $f$ or $\bar{f}$ is holomorphic in $\Omega$ .

Problem 2.11.3 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.11 "Harmonic Functions"

If $u$ is a harmonic function in a region $\Omega$ , what can you say about the set of points at which the gradient of $u$ is 0? (This is the set on which $u_{x} = u_{y} = 0$ ).

Problem 2.11.5 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.11 "Harmonic Functions"

Suppose $f \in H(\Omega )$ and $f$ has no zero in $\Omega$ . Prove that $\log \left|f\right|$ is harmonic in $\Omega$ , by computing its Laplacian. Is there an easier way?

Problem 2.11.7 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.11 "Harmonic Functions"

(a)(b)

(a)

If $f \in H(\Omega ), f(z) \neq 0$ for $z \in \Omega$ , and $-\infty < \alpha < \infty$ , prove that

\Delta \left(\left|f\right|^{\alpha }\right) = \alpha ^{2}\left|f\right|^{\alpha -2}\left|f^{\prime }\right|^{2}

by proving the formula

\partial \bar{\partial }(\psi \circ (\overline{f f})) = \left(\varphi \circ \left|f\right|^{2}\right) \cdot \left|f^{\prime }\right|^{2}

in which $\psi$ is twice differentiable on $(0, \infty )$ and

\varphi (t) = t \psi ^{\prime \prime }(t)+\psi ^{\prime }(t)

(b)

Assume $f \in H(\Omega )$ and $\Phi$ is a complex function with domain $f(\Omega )$ , which has continuous second-order partial derivatives. Prove that

\Delta [\Phi \circ f] = [(\Delta \Phi ) \circ f] \cdot \left|f^{\prime }\right|^{2}

Show that this specializes to the result of $(a)$ if $\Phi (w) = \Phi (\left|w\right|)$ .

Problem 5.10 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"

Let $u, v$ be a pair of conjugate harmonic functions in a domain $G$ , and let $\xi , \eta$ be a pair of conjugate harmonic functions in a domain $D$ . Let, for every $z = x+i y \in G$ , the value $u(x, y)+i v(x, y)$ belong to the domain $D$ . Prove that the pair of functions $U, V$ of the form

\begin{aligned} U(x, y) & = \xi (u(x, y), v(x, y)) \\ V(x, y) & = \eta (u(x, y), v(x, y)) \end{aligned}

forms a pair of conjugate harmonic functions in the domain $G$ .

Problem 5.11 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"

Let $u, v$ be a pair of conjugate harmonic functions in a domain $G$ , and let the functions $u$ and $v$ not vanish simultaneously at any point of the domain $G$ . Prove that the function

U(x, y) = \ln \left[u^{2}(x, y)+v^{2}(x, y)\right]

is harmonic in the domain $G$ .

Problem 5.15 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)

Let $P(x, y)$ be a polynomial in $x$ and $y$ with complex coefficients. Denote

P^{*}(z, \bar{z}) = P\left(\frac{z+\bar{z}}{2}, \frac{z-\bar{z}}{2 i}\right)

Prove that

(1)

the functions $u(x, y) = \mathrm{Re}\left(P(x, y)\right), v(x, y) = \mathrm{Im}\left(P(x, y)\right)$ satisfy the Cauchy--Riemann conditions if and only if the polynomial $P^{*}(z, \bar{z})$ does not depend on $\bar{z} ;$

(2)

the polynomial $P(x, y)$ satisfies Laplace's equation $\Delta P = 0$ if and only if the polynomial $P^{*}(z, \bar{z})$ can be represented in the form $Q_{1}(z)+Q_{2}(\bar{z})$ , where $Q_{1}$ and $Q_{2}$ are polynomials.

References

[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.

[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.