# Jackson Green function expansion in spherical coordinates

In Jackson's "Classical Electrodynamics" third edition in section 3.9 page 121 while explaining a Green function expansion the following equation is attained:

$$\frac{1}{r}\frac{d^2}{dr^2}(rg_l(r,r')) - \frac{l(l+1)}{r^2}g_l(r,r')=-4π\delta(r-r')$$ $$(eq.1)$$

which for the relevant boundary conditions has as the solution

$$C(r_<^l-\frac{a^{2l+1}}{r_<^{l+1}})(\frac{1}{r_>^{l+1}}-\frac{r_>^{l}}{b^{2l+1}})$$ $$(eq2)$$

then to obtain the constant $$C$$ he multiplies $$eq1$$ by $$r$$ and integrates from $$r'-\epsilon$$ to $$r'+\epsilon$$ but for some reason completely leaves out the $$2$$nd term as if it integrates to zero or something... This has been bothering me for a while since I had to use this method a couple of times in the exercises and often I had to ignore some term but I don't understand why this is the case...

something i thought of while writing: maybe to get the constant I just have to consider the $$l=0$$ case or something? but then why are there $$l$$s in the solution?

The second term is bounded in the region of integration. As we are integrating over a region of length $$2\epsilon$$ it is less in magnitude than $$2M\epsilon$$, where $$M$$ is the bound. This becomes negligible as $$\epsilon\to 0$$. The other terms are are singular and give contributions that do not go to zero with epsilon.
• First of all thanks for the answer... but I still don't understand. What do you mean with the bound $M$? Commented Nov 11, 2019 at 14:23
• I mean that there exists some positive number $M$ such that everywhere in the interval $r'-\epsilon$ to $r'+\epsilon$ we have $|(l(l+1)/r^2) g_l(r,r')|<M$. Hence $$\int_{r'-\epsilon}^{r'+\epsilon}(l(l+1)/r^2) g_l(r,r')dr<2M\epsilon$$. Commented Nov 11, 2019 at 19:31
• Oh, ok, and the first term is infinite because there is a discontinuity in $g(r,r')$ and thus cancels the $\epsilon$? Commented Nov 11, 2019 at 19:44