in Fluid Mechanics by Landau and Lifschitz (a fairly well-known book - I'm just mentioning it for those who might have it) they are discussing
"As we know, Laplace's equation has a solution l/r, where r is the distance from the origin. The gradient and higher space derivatives of 1/r are also solutions. All these solutions, and any linear combination of them, vanish at infinity. Hence the general form of the required solution of Laplace's equation at great distances from (a contour enclosing the origin) is
$$ \Phi(r) = -a/r + A*Grad(1/r)+....\, $$
(A is a vector)
where a and A are independent of the coordinates; the omitted terms contain higher-order derivatives of 1/r..."
I understand how the 1/r is a solution of Laplace's equation, but fail to see how the Grad(1/r) is also a solution (not to mention the whole 'higher space derivatives' thing)
Can anyone point me to a proof of the gradient being a solution (assuming that we're in 3 space dimensions) - if not showing how to arrive at the more general conclusion? I can't figure out where I'm going wrong in my calcs (and it's driving me a bit nutty....)
It does occur to me that perhaps they are only speaking of a Laurent expansion at large distances from the origin and not actually claiming that the "A*Grad(1/r)" is any type of a solution to Laplace's equation in 3D - can someone confirm this?