TOPICS
Search

Green's Function--Poisson's Equation


Poisson's equation is

 del ^2phi=4pirho,
(1)

where phi is often called a potential function and rho a density function, so the differential operator in this case is L^~=del ^2. As usual, we are looking for a Green's function G(r_1,r_2) such that

 del ^2G(r_1,r_2)=delta^3(r_1-r_2).
(2)

But from Laplacian,

 del ^2(1/(|r-r^'|))=-4pidelta^3(r-r^'),
(3)

so

 G(r,r^')=-1/(4pi|r-r^'|),
(4)

and the solution is

 phi(r)=intG(r,r^')[4pirho(r^')]d^3r^'=-int(rho(r^')d^3r^')/(|r-r^'|).
(5)

Expanding G(r_1,r_2) in the spherical harmonics Y_l^m gives

 G(r_1,r_2)=sum_(l=0)^inftysum_(m=-l)^l1/(2l+1)(r_<^l)/(r_>^(l+1))Y_l^m(theta_1,phi_1)Y^__l^m(theta_2,phi_2),
(6)

where r_< and r_> are greater than/less than symbols. this expression simplifies to

 g(r_1,r_2)=1/(4pi)sum_(l=0)^infty(r_<^l)/(r_>^(l+1))p_l(cosgamma),
(7)

where p_l are Legendre polynomials, and cosgamma=r_1·r_2. Equations (6) and (7) give the addition theorem for Legendre polynomials.

In cylindrical coordinates, the Green's function is much more complicated,

 G(r_1,r_2)=1/(2pi^2)sum_(m=-infty)^inftyint_0^inftyI_m(krho_<)K_m(krho_>)e^(im(phi_1-phi_2))cos[k(z_1-z_2)]dk,
(8)

where I_m(x) and K_m(x) are modified Bessel functions of the first and second kinds (Arfken 1985).


Explore with Wolfram|Alpha

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 485-486, 905, and 912, 1985.

Cite this as:

Weisstein, Eric W. "Green's Function--Poisson's Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GreensFunctionPoissonsEquation.html

Subject classifications