Greens Function or Fundamental Solution

The final result of Example 10.1 is Green s second identity for two vectors defined by f = (/ VT and g = 7 Vc/l The only aspect that remains to be resolved is a correct selection of the extra function / . The best selection is a function that satisfies a special form of Poisson s equation given by 

Additionally, we can now make use of two useful properties of the delta function, 

Basically, when integrating the product of a function and the Dirac delta function, the Dirac delta function acts like a filter, resulting in the value of that function evaluated at the point where the Dirac delta function is applied. 

The most common technique for the derivation of fundamental solutions is to use integral transforms, such as, Fourier, Laplace or Hankel transforms [29, 39]. For simple operators, such as the Laplacian, direct integration and the use of the properties of the Dirac delta are typically used to construct the fundamental solution. For the case of a two-dimensional Laplace equation we can use a two-dimensional Fourier transform, F, to get the fundamental solution as follows, 

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