Case 2: $B^{(0)}(\mathbf{x}) = (1,0,0)$

 

$$F(\mathbf{x}';\mathbf{x}) = \iiint \frac{\partial^2 G}{\partial x' \partial z'} \, dx' \, dy' \, dz'$$

$$= \int G \, dy'$$

$$= \frac{-1\,}{4\pi} \mathrm{sinh}^{-1}\left(\frac{y'}{\sqrt{{x'}^2 + {z'}^2}}\right)$$ (23)

using equation 29 from Appendix A.

Again, this can be substituted into equations 14 and 21 to get $B^{(1)}_z$.