Cauchy and the (other) mean value theorem.

I just realised that there's a neat relationship between Cauchy's integral formula and the mean value theorem for harmonic functions of two variables, so for the benefit of anybody else who hasn't already noticed it, here it is.

First, let's remember what the Cauchy integral formula tells us.

If \(f\) is a holomorphic function on a simply connected domain, \(D\), \(a \in D\), and \(C\) is a simple closed contour in \(D\) surrounding \(a\), then \[ f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-a} dz. \]

I confess, this always seems like magic to me.

One way of seeing that it is true is to note that for a holomorphic function continuous deformation of the countour doesn't change the integral: therefore we can take \(C\) to be an extremely small circle centred on \(a\), so that \(f(z)\) is very close to \(f(a)\) all around the contour. Then if we parameterise \(C\) as \(a+\varepsilon e^{i \theta}\), where \(0 \leq \theta < 2\pi\), the integral is approximately \[ \frac{f(a)}{2 \pi i} \oint_C \frac{1}{z-a} dz = \frac{f(a)}{2 \pi i} \int_0^{2 \pi} i d\theta = f(a) \] and the approximation can be made arbitrarily good by choosing \(\varepsilon\) small enough.

And even with that, it still seems like magic.

Now, let's take a look at this in terms of real and imaginary parts. We can think of the complex number \(z\) as \(x+iy\), and then thinking about \(f\) as a function of \(x\) and \(y\). As is customary, I write: \[ f(z) = u(x,y)+iv(x,y) \] where \(f\) being holomorphic tells us that \(u\) and \(v\) satisfy the Cauchy-Riemann equations, \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} \] and as a consequence \[ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} = 0 = \frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2} \] so that both \(u\) and \(v\) are harmonic functions.

It's less obvious, but equally true that if \(u\) is a harmonic function of \(x\) and \(y\), then there is another harmonic function \(v\) such that \[ f(z)=f(x+iy) = u(x,y) + iv(x,y) \] is holomorphic.

So now let's see what the Cuachy integral formula says, in terms of \(u\) and \(v\).

To this end, let \(C\) be a circular contour of radius \(R\), centred on \(a\), and parameterised by \(\theta\) as \[ a+Re^{i\theta}, \qquad 0 \leq \theta < 2\pi \]

Then we have \[ \begin{split} \frac{1}{2 \pi i} \oint_C \frac{f(z)}{z-a}dz &= \frac{1}{2 \pi i} \int_0^{2 \pi} \frac{f(z(\theta))}{Re^{i\theta}} iRe^{i\theta} d\theta \\ &= \frac{1}{2\pi} \int_0^{2\pi} u(x(\theta),y(\theta))d\theta + i \frac{1}{2\pi} \int_0^{2\pi} v(x(\theta),y(\theta))d\theta \end{split} \]

So the real part of \(f(a)\) is the mean value of \(u(x,y)\) on a(ny) circle centred on \(a\), and the imaginary part is the mean value of \(v(x,y)\) on a(ny) circle centred on \(a\).

But the real part of \(f(a)\) is just \(u(Re(a),Im(a))\), and similarly for \(v\)!

This result, that the value of \(u(x,y)\) for harmonic \(u\) is the mean value of \(u\) on any circle centred on \((x,y)\), is the mean value theorem for harmonic functions.

We can go backwards: assuming the mean value theorem for harmonic functions, we can deduce the Cauchy integral formula (at least, if we are happy that contour integrals of holomorphic functions are unchanged by a deformation of the contour).

So it turns out that Cauchy's integral formula, a fundamental result in complex analysis, is equivalent to a real analysis result about the properties of harmonic functions.

In retrospect, I suppose it should be obvious in some sense that Cauchy's integral formula relies on the properties of harmonic functions, since holomorphic functions are built out of harmonic ones. But I hadn't put the two ideas together like this before.