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.

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