9 Green’s Functions and Images

The fundamental solution solves Poisson’s equation in all of space. In a bounded region, the boundary must be taken into account, and the tool is the Green’s function: the response at \(y\) to a unit point source at \(x\), with the boundary condition built in. Once it is known, every Dirichlet problem in the region is solved by one integral. George Green introduced it in 1828, in a self-published essay on electricity that went almost unread until William Thomson found it seventeen years later; Thomson then found the Green’s functions of the half-space and the ball by the method of images, which this section develops.

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