Harmonic Equations

In an electrical vacuum, the electrical potential obeys Laplace's equation:

$$\frac{\partial^2V}{\partial x^2} + \frac{\partial^2V}{\partial y^2}=0$$ (1.1)

A real-valued function is considered harmonic in a domain D if all of its second-order partial derivatives are continuous in D, and if at each point in D the function satisfies Laplace's equation [1]. Such functions come from the real and imaginary parts of complex analytical functions.

Consider the function:

$$f(z)=u(x,y)+iv(x,y)$$ (1.2)

We assume the derivative of this function exists at ${\rm z_o=x_o + iy_o}$. We let $\Delta z = \Delta x+i\Delta y \rightarrow 0$ along the x and y axis independently, which is to say,

$$\frac{df(z_o)}{dz} = \lim_{\Delta z = \Delta x \rightarrow 0}\lef... ...x_0 + \Delta x, y_0)}{\Delta x}-\frac{u(x_0,y_0)+iv(x_0,y_0)}{\Delta x}\right)$$

Likewise for ${\rm\Delta z=i\Delta y}$, such that we have the set of equations,

$$\frac{df(z_0)}{z}=\frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}=-i\frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}$$ (1.3)

These are better known as the Cauchy-Riemann equations,

$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}$$ (1.4)

The demand that

$$\frac{\partial}{\partial y}\frac{\partial u}{\partial x}= \frac{\partial}{\partial x}\frac{\partial u}{\partial y}$$

gives

$$\frac{\partial^2v}{\partial y^2}=-\frac{\partial^2v}{\partial x^2}$$

Which is simply Laplace's equation, establishing our claim that potential functions are components of complex functions. With this fact comes the many useful tools of complex analysis. Specifically, Cauchy's Integral Formula reads,

$$f(z_0)=\frac{1}{2\pi}\oint \frac{f(z)}{z-z_0}dz$$ (1.5)

for some contour. If we parameterize that contour with $z = z_0 + Re^{it}, 0 \leq t \leq 2\pi$, and

$$f(z)dz = f(z)\frac{dz}{dt}dt$$

This gives

$$f(z_0)=\frac{1}{2\pi i}\int^{2\pi}_0\frac{f(z_0+Re^{it})}{Re^{it}}iRe^{it}dt$$

This is better known as the Mean-Value property,

$$f(z_0)=\frac{1}{2\pi}\int^{2\pi}_0f(z_0+Re^{it})dt$$ (1.6)

A harmonic function evaluated at some point is equal to the average value of that function around some circle (or sphere) centered at that point. The Jacobi method follows directly from this.