A further identity follows for differential operators that are scalar polynomials of the derivative, L = P N ( ∂ ∂ x ) . The fundamental theorem of algebra, combined with the fact that ∂ ∂ x commutes with itself, guarantees that the polynomial can be factored, putting L in the form: L = ∏ ∏ i = 1 N ( ∂ ∂ x − − z i ) , where z i are the zeros of P N ( z ) . Taking the Fourier transform of L G ( x , s ) = δ δ ( x − − s ) with respect to both x and s gives: G ^ ^ ( k x , k s ) = δ δ ( k x − − k s ) ∏ ∏ i = 1 N ( i k x − − z i ) . The fraction can then be split into a sum using a partial fraction decomposition before Fourier transforming back to x and s space. This process yields identities that relate integrals of Green's functions and sums of the same. For example, if L = ( ∂ ∂ x + γ γ ) ( ∂ ∂ x + α α ) 2 then one form for its Green's function is: G ( x , s ) = 1 ( γ γ − − α α ) 2 Θ Θ ( x − − s ) e − − γ γ ( x − − s ) − − 1 ( γ γ − − α α ) 2 Θ Θ ( x − − s ) e − − α α ( x − − s ) + 1 γ γ − − α α Θ Θ ( x − − s ) ( x − − s ) e − − α α ( x − − s ) = ∫ ∫ Θ Θ ( x − − s 1 ) ( x − − s 1 ) e − − α α ( x − − s 1 ) Θ Θ ( s 1 − − s ) e − − γ γ ( s 1 − − s ) d s 1 . While the example presented is tractable analytically, it illustrates a process that works when the integral is not trivial (for example, when ∇ ∇ 2 is the operator in the polynomial).