8.1 Harmonic functions and the fundamental solution

Examples in the plane are \(x^2 - y^2\), \(e^x\cos y\), \(r^k\cos k\theta \) and \(r^k\sin k\theta \), the real and imaginary parts of \(z^2\), \(e^z\) and \(z^k\). A harmonic function that depends only on \(r = |x|\) satisfies \(u'' + \frac {n - 1}ru' = 0\), so \(u' = Cr^{1 - n}\), giving \(a + b\log r\) in the plane and \(a + b/r\) in space. Normalised so that \(-\Delta \Phi = \delta \), the point source, these are the fundamental solutions \[ \Phi (x) = -\frac 1{2\pi }\log |x| \quad (n = 2), \qquad \Phi (x) = \frac 1{4\pi |x|} \quad (n = 3) , \] the potentials of a line charge and of a point charge. They are harmonic except at the origin, and the normalisation means that the flux of \(-\nabla \Phi \) through any sphere about the origin is \(1\): in three dimensions, \(-\partial _r\Phi = 1/4\pi r^2\) times the area \(4\pi r^2\). The solution of Poisson’s equation in all of space is the superposition \(u = \Phi * f\), as the next section, on Green’s functions, will show.

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