Lecture notes — Green's function

Green's function definition: G satisfies LG=delta. For a linear differential operator L, the Green's function G(x; s) is the response at x to a unit impulse at s. Formally, LG(x; s) = delta(x - s).

Once G is known, the solution to L u = f is u(x) = integral G(x; s) f(s) ds.

Common cases:
- Laplace on a half-plane → method of images.
- 1D Sturm–Liouville → modified Bessel functions.
