A standard argument shows that the solution to (1) is written as
where G is the fundamental solution of Laplace's equation given by
and is the crack opening displacement defined by
The unknown functions u on and on S are determined as the solutions to the following integral equations:
where the integration symbol with a superimposed = on the RHS of (4) stands for the finite part. As a matter of fact the hypersingular integral equation in (4) is obtained as one computes the normal derivative of (2) in and takes the limit onto S. A simple but rigorous proof that the limit takes the form in (4) is found for example in Nishimura & Kobayashi.