Dubious Taylor Expansion - Jackson Classical Electrodynamics pg. 35

[hachensach]

For details, I posted a pdf at http://www.scribd.com/doc/26127683/Jackson-35

To sum up, on pg. 35, Section 1.7 - Poisson and Laplace Equations, of Jackson's Classical Electrodynamics, 3rd Edition, the author allegedly expresses the charge density rho(x',y',z') as a Taylor series around (x',y',z') = (x,y,z). He writes

rho(x',y',z') = rho(x,y,z) + (r^2/6)*nabla^2 rho + ...

How'd he get this? It doesn't seem to follow from the vector form of Taylor series for a 3-variable function.

He follows this by an equation which contains the "big-oh" expression

O(a^2, (a^2)*log a)

This equation was allegedly obtained by "direct integration". How? And what does the above big-oh expression mean?

[Clem]

Jackson's Chapter I is a worthy introduction to what will follow, but those equations are almost correct, if inscrutable. For his Taylor's expansion, he has integrated the full expansion over solid angle. This gets rid of the first order derivatives.
As a misprint (I hope) he has left out the zeroth order term in his unnumbered equation at the bottom of p. 39. This is in his (better) 2nd Edition, which is all I have in hand at the moment. Then nontrivial angular integration over the (x.del)(x.del) does lead to the equation you quote. The same is true for the integration of the zeroth and second order Taylor terms. To integrate a^2(r^2+a^2)^-(5/2), successively add and subtract a^2 to the numerator, and then let r=a tan\theta. Good luck as needed.

The "big-oh expression" means that there is a term that depends on a^2 and another one that depends on a^2 log a. You will get those with the integration. It does not matter what they are since they go to zero in the llinmit.
If you are self-studying ED, I could recommend other books.