Formula sheet PDE 401

Laplace operator
The Laplace operator del squared phi equals the sum of second partial derivatives. A function satisfying del squared phi equals zero is called harmonic.

Green function for Laplacian
On a domain Omega with boundary, the Green function G of x y satisfies
del squared G of x y equals minus delta of x minus y inside Omega
G of x y equals zero on the boundary of Omega

Poisson kernel on the disk
P r theta minus phi equals one minus r squared over two pi times one minus two r cos of theta minus phi plus r squared

Wave equation general solution one dimension
u of x t equals f of x minus c t plus g of x plus c t
