Integral Formulation of Poissons Equation

We can now generate an equivalent integral formulation for Poisson s equation 

The integral formulation for Poisson s equation is found the same way as for Laplace s equation (using Green s second identity, Theorem (10.1.3)), except that now the second volume integral is kept in Green s second identity. For a point xq V the integral formulation 

In the mathematical literature this equation is interpreted in the following way  

The use of this terminology reflects the fact that, as the point x0 approaches the surface of the domain, the double-layer potential has a discontinuity and it must be taken it into account by multiplying the field w(x0) by a coefficient. 

In terms of heat transfer this can be physically interpreted that, if the infinite heat source is at the boundary (the infinite character is given by the delta function), then for a smooth surface only half of the delta function must be included. The general integral representation of Poisson s equation becomes, 

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