Question

legendre polynomial explanation

Answer 1

POTENTIAL OF r = k/r =k/(r^2 + 2r*vt + v^2xt^2)^0.5 = (k/r)(1+2r*vt/r^2 + v^2xt^2)^(-0.5) = (k/r)(1+nx + n(n-1)x^2/2! + n(n-1)(n-2)x^3/3! + ....

x = 2r*vt/r^2 + v^2xt^2, n = -0.5, r*vt = abs(r)xabs(v)xcos_theta x t

if you use this relation, you will find the dipole, quadrupole, .... and
the time delay equation, ~ k/(r-r*vt)

Answer 2

there is some mistakes of algebra, sorry. but the eccential idea is in geometric. so you can derive the above equation and the mistakes.(ex, r = distance, vt = distance, and cos_theta = angle of two vectors of r, vt