# The fundamental solution for Laplace's equation in cylindrical coordinates

I am working my way through the excellent textbook of Garabedian on partial differential equations and have two questions related to the topic of the fundamental solution in chapter 5:

1. Garabedian defines a fundamental solution of an elliptic equation $$u_{xx}+u_{yy}+au_x+bu_y+cu=0$$ to be a solution on the form $$S=S(x,y;\xi,\eta)=A(x,y;\xi,\eta)\log\frac{1}{r}+B(x,y;\xi,\eta).$$ He points out that the fundamental solution becomes logarithmically infinite when $$x\rightarrow\xi,\ y\rightarrow\eta .$$ He then writes: "It is the simplest conceivable singular solution." My question is: In what sense ?
2. Then, in the associated exercise 7, he is considering a particular fundamental solution, which represents the gravitational potential of a homogeneous circular wire of radius $$\rho$$. Part of the exercise is to show that it can be expressed as $$S(z,r;\zeta,\rho)=\frac{\rho}{Q_1}\int^{\infty}_{-\infty}\frac{dt}{\sqrt{(1+t^2)(k^2+t^2)}}$$ where $$k=\frac{P_1}{Q_1}=\sqrt{\frac{(z-\zeta)^2+(r-\rho)^2}{(z-\zeta)^2+(r+\rho)^2}},$$ $$P_1$$ and $$Q_1$$ being the shortest and longest distance, respectively, from the observer point to the wire. However, Garabedian next asks: "Establish that the Riemann function $$A(z,r;\zeta,\rho)$$ [for the above fundamental solution] can be identified, except a factor of $$-2\pi i$$. as the period of the term on the right corresponding to an increment of $$2\pi$$ in the argument of the parameter $$k$$." I believe that I (sort of) understand operationally what is asked for, but I would be grateful for some hints on how to proceed.