Harmonic Functions

A real-valued function $\varphi (x,y)$ is said to be harmonic in a domain $D$ if its second partial derivatives are continuous in $D$ and if at each point of $D$, $\varphi$ satisfies Laplace's equation

$$\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\varphi }{\partial y^2}=0$$

For an analytic function $f(x,y)=u(x,y)+v(x,y)i$, the second partial derivatives of $u$ and $x$ satisfy Laplace's equation. Therefore, we have the following theorem:

The real and imaginary parts of a function analytic in a domain $D$ are harmonic in $D$.

When $u(x,y)$ is harmonic in a domain $D$, a function $v(x,y)$ such that $u+vi$ is analytic in $D$ is called a harmonic conjugate of $u(x,y)$. This means that given a harmonic function $u(x,y)$, we can always find a harmonic conjugate $v(x,y)$ (this is not true for all domains though.)

Given a harmonic function $u(x,y)$ on a suitable domain, we can use the Cauchy-Riemann equations to find a harmonic conjugate $v(x,y)$, as follows.

From the Cauchy-Riemann equations we have $u_x=v_y$ (using the subscript notation for partial derivatives). Then,

$$\begin{array}{cc}v(x,y)=\int u_{x}dy=\int v_{y}dy& \text{(a)}\end{array}$$

is the function we're looking for, except the constant of integration will be some unknown $h(x)$.

We'll say then that

$$\begin{array}{cc}v(x,y)=g(x,y)+h(x),& \text{(b)}\end{array}$$

where $g(x,y)$ will explicitly be the antiderivative of $v_y$ with respect to $y$, but $h(x)$ will be an unknown function of $x$.

Now, we can take

$$\begin{array}{cc}\frac{\partial }{\partial x}(g(x,y)+h(x))=v_x=g^{\prime }(x,y)+h^{\prime }(x,y).& \text{(c)}\end{array}$$

From the Cauchy-Riemann equations, $v_x=-u_y$. Comparing $g^{\prime }(x,y)+h^{\prime }(x)$ to $-u_y$, we find $h^{\prime }(x)$ as the missing terms from $g^{\prime }(x,y)$ in $-u_y$, if any. Integrating these gives the value of $h(x)$, which we can plug into (b) to get $v(x,y)$.

Note that this is the same mechnical procedure we use to solve first order exact differential equations.

We can use this to find analytic functions from a harmonic function by using the harmonic conjugates and the harmonic function as the real and imaginary parts of a complex function.

One interesting property of harmonic functions and their conjugates is that they define orthogonal families of curves - all of their intersections are at right angles to each other.