Delta-function method

A method for finding the Green function of a linear differential equation in mathematical physics (i.e. a method for determining the function of the effect of a point source) with the aid of the delta-function $\delta(x)$. The Green function $G(x,x')$ of a linear differential operator $L$ is defined by the equation

$$L(x)G(x,x')=\delta(x-x'),$$

or $G(x,x')=-L^{-1}(x)\delta(x-x')$, i.e. it expresses the effect of a point source located at the point $x'$ on the value of the resulting perturbation at the point $x$. The form of the inverse operator $L^{-1}$ is most simply determined in the frequently occurring case when $L$ is a differential operator with constant coefficients (independent of $x$). The solution of a non-homogeneous linear differential equation of a general type for a perturbation $\phi$ with source $\rho$,

$$L(x)\phi(x)=-\rho(x),$$

is described with the aid of the Green function $G(x,x')$ as the convolution

$$\phi(x)=\int G(x,x')\rho(x')\,dx',$$

the integration being effected over the entire domain of action of the source $\rho$.

References

[1]	D. Ivanenko, A. Sokolov, "Classical field theory" , Mosow-Leningrad (1951) (In Russian)

References

[a1]	P.M. Morse, H. Feshbach, "Methods of theoretical physics" , 1–2 , McGraw-Hill (1953)