Complex Analysis

Lecture 25

Harmonic functions

Remark: \(v\) is harmonic conjugate of \(u\) \(\nRightarrow\) \(u\) is a harmonic conjugate of \(v\).

Example: \(u(x,y) = x^2-y^2\),  \(v(x,y) = 2xy\). Since \(u+iv = z^2\) is entire \(\implies\) \(v\) is a harmonic conjugate of \(u\).

But if \(u\) were a harmonic conjugate of \(v\), then \(g = v+iu\) would be analytic: Check via C/R, nowhere analytic.

Remark: Suppose \(u\) is harmonic on a simple connected domain \(\Omega\). Then \(u\) has a harmonic conjugate on \(\Omega\).

Physical problems

"Physical" configurations often modelled by solutions of partial differential equations (PDE).

Generally, interested in solving a PDE subject to associated initial/boundary conditions.

Dirichlet problem

\( \text{(D) } \left\{ \begin{array}{cc} \Delta u = 0 & \text{in } \Omega\\ u|_{\partial \Omega} = \varphi & (*) \end{array} \right. \)

\((*)\) Says \(u(\mathbf x)= \varphi(\mathbf x)\) for all \(\mathbf x \in \partial \Omega\)

       (\( \varphi: \partial \Omega \ra \R\)).

\(\Omega, \varphi\) are known/given, and \(u\) is the unknown.

\(\text{(D)}\) is called the Dirichlet problem for Laplace's equation, a. k. a. boundary problem of the first kind.

Dirichlet problem

One way to "solve" \(\text{(D)}\) is to find \(u\) that minimises \[\int_{\Omega} \left|\nabla u\right|^2 d\mathbf x\] subject to \(u|_{\partial \Omega} = \varphi\).

Neumann problem

Also important: Boundary conditions of the second kind, called Neumann boundary conditions

\( \text{(N) } \left\{ \begin{array}{cc} \Delta u = 0 & \\ \ds\frac{\partial u}{\partial \mathbf \nu} = \Psi & \text{on } \partial \Omega \end{array} \right. \)

In practice often have homogeneous Neumann boundary conditions, i. e., \(\Psi = 0\).

Transformation of harmonic functions

Theorem: If \(f\) is conformal and \(h\) is harmonic in \(\Lambda\), then \(H\) is harmonic in \(\Omega\), where \[H(x,y) = h\left(u(x,y), v(x,y)\right).\]

Transformation of harmonic functions

Proof: Mesy in general, but straightforward if \(\Lambda\) is simply connected.

Example: \(h(x,y) = e^{-v}\sin u\) is harmonic in UHP (upper half plane).

Define \(w=x^2\) on \(\Omega =\) 1st quadrant.

Transformation of harmonic functions

Example (cont): So \(u = x^2 -y^2\) and \( v = 2xy\).

The previous Theorem implies \[H(x,y) = e^{-2xy}\sin \left(x^2-y^2\right). \] is harmonic on \(\Omega\).