Two- and Three-Dimensional Green’s Functions

We discussed Green’s function briefly (Section 7.2). The one-dimensional Green’s function of the Laplace equation, with Dirichlet boundary conditions, is the solution of the differential equation 7A.1a d 2 G d x 2 = − δ x − x ′ , 0 < x < L , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn7a_1a.jpg"/> subject to the boundary conditions 7A.1b G = 0 , x = 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn7a_1b.jpg"/> 7A.1c G = 0 , x = L , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn7a_1c.jpg"/> and is shown to be 7A.2a G = G 1 = x L − x ′ L , 0 < x < x ′ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn7a_2a.jpg"/> 7A.2b G = G 2 = x ′ L − x L , x ′ < x < L . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn7a_2b.jpg"/>