>>12999341consider e.g. the Poisson equation . since it's linear, there is the superposition principle which basically says that we can do this:
1. write as a sum
2. solve , call the solution (of course the point is to choose such that we can solve the equation)
3. now solves the original equation
the idea now is to mimic this approach, but replace the finite sum by an "uncountable sum" (which is just integration), and write the function as the uncountable sum of its individual values using dirac delta
1. write . this is supposed to say that if is (informally) a function which is zero everywhere except at where it's , then is the continuous sum of these functions ranging over the whole domain of .
2. solve the equation for each "summand", i.e. solve . call this solution . THIS IS WHAT THE GREEN FUNCTION IS.
3. the solution to is the sum of the individual solutions from the second step. of course here sum ranging over all is again integration over . thus the formula is .
