1.
Show that all partial derivatives of a harmonic function are harmonic.
Exercises 6.3 Exercises
2.
Suppose \(u(x,y)\) and \(v(x,y)\) are harmonic in \(G\text{,}\) and \(c \in \R\text{.}\) Prove that \(u(x,y) + c \, v(x,y)\) is also harmonic in \(G\text{.}\)
3.
Give an example that shows that the product of two harmonic functions is not necessarily harmonic.
4.
Let \(u(x,y) = e^x \sin y\text{.}\)
(a)
(b)
5.
Consider \(u(x,y) = \ln \left( x^2 + y^2 \right)\text{.}\)
(a)
(b)
Prove that \(u\) is not the real part of a function that is holomorphic in \(\C
\setminus \{ 0 \}\text{.}\)
6.
Show that, if \(f\) is holomorphic and nonzero in \(G\text{,}\) then \(\ln|f(x,y)|\) is harmonic in \(G\text{.}\)
7.
Suppose \(u(x,y)\) is a function \(\R^2 \to \R\) that depends only on \(x\text{.}\) When is \(u\) harmonic?
8.
9.
Suppose \(f\) is holomorphic in the region \(G \subseteq \C\) with image \(H := \left\{ f(z) : \, z
\in G \right\}\text{,}\) and \(u\) is harmonic on \(H\text{.}\) Show that \(u(f(z))\) is harmonic on \(G\text{.}\)
10.
(a)
Show that the Laplace equation for \(u(r, \phi)\) is
\begin{equation*}
\frac 1 r \, u_r + u_{ rr } + \frac 1 {r^2} u_{ \phi
\phi } \ = \ 0 \,\text{.}
\end{equation*}
(b)
(c)
(d)
11.
Hint.
12.
Suppose \(u(x,y)\) is a harmonic polynomial in \(x\) and \(y\text{.}\) Prove that the harmonic conjugate of \(u\) is also a polynomial in \(x\) and \(y\text{.}\)
13.
Recall from Exercise 4.5.31 the Poisson kernel
\begin{equation*}
P_r(\phi) \ = \ \frac{ 1-r^2 }{ 1 - 2r \cos(\phi) + r^2 } \,\text{,}
\end{equation*}
where \(0 \lt r \lt 1\text{.}\) In this exercise, we will prove the Poisson Integral Formula: if \(u\) is harmonic on an open set containing the closed unit disk \(\overline D[0,1]\) then for any \(r\lt 1\)
\begin{equation}
u \! \left( r \, e^{ i \phi } \right) \ = \ \frac{ 1 }{ 2 \pi }
\int_0^{ 2 \pi } u \! \left( e^{ i t } \right) P_r(\phi - t)
\, \diff{t} \, \text{.}\tag{6.2}
\end{equation}
Suppose \(u\) is harmonic on an open set containing \(\overline D[0,1]\text{.}\) By Exercise 6.3.14 we can find \(R_0>1\) so that \(u\) is harmonic in \(D[0,R_0]\text{.}\)
(a)
Recall the Möbius function
\begin{equation*}
f_a(z) \ = \ \frac{ z-a }{ 1 - \conj{a} z } \,\text{,}
\end{equation*}
for some fixed \(a \in \C\) with \(|a| \lt 1\text{,}\) from Exercise 3.6.9. Show that \(u(f_{ -a }(z))\) is harmonic on an open disk \(D[0,R_1]\) containing \(\overline D[0,1]\text{.}\)
(b)
\begin{equation}
u(a) \ = \ \frac{ 1 }{ 2 \pi i } \int_{ C[0,1] } \frac{
u(f_{ -a }(z)) }{ z } \, \diff{z} \,\text{.}\tag{6.3}
\end{equation}
(c)
Recalling, again from Exercise 3.6.9, that \(f_a(z)\) maps the unit circle to itself, apply a change of variables to (6.3) to prove
\begin{equation*}
u(a) \ = \ \frac{ 1 }{ 2 \pi } \int_0^{ 2 \pi } u \!
\left( e^{ i t } \right) \frac{ 1 - |a|^2 }{ \left|
e^{ i t } - a \right|^2 } \, \diff{t} \,\text{.}
\end{equation*}
(d)
14.
Suppose \(G\) is open and \(\overline D[a,r] \subset G\text{.}\) Show that there is \(R>r\) so that \(\overline D[a,r] \subset D[a,R] \subset
G\text{.}\)
Hint.
If \(G=\C\) just take \(R=r+1\text{.}\) Otherwise choose some \(w\in \C\setminus G\text{,}\) let \(M=|w-a|\text{,}\) and let \(K=\overline D[a,M] \setminus G\text{.}\) Show that \(K\) is nonempty, closed and bounded, and apply Theorem A.0.1 to find a point \(z_0\in K\) that minimizes \(f(z)=|z-a|\) on \(K\text{.}\) Show that \(R=|z_0-a|\) works.
