Post by Admin on Dec 13, 2019 9:48:00 GMT
Unless your copy of the book is the 2000 printing (with corrections) or a later printing, the statement of this exercise contaims errors.
The version in the 2000 printing contains the following corrected version.
Use the idea behind the method of images to show that if $0<\text{Re}(p)<1$ then
the Green's function of the half-disc $\text{Re}(z)\geq 0,|z|\leq 1$ is
$\displaystyle\mathcal{G}_p(z)=-\ln\bigg|\frac{z-p}{\overline{p}z-1}\bigg|+\ln\bigg|\frac{z+\overline{p}}{pz+1}\bigg|$ .
Check this by getting the computer to draw the level curves of $\mathcal{G}_p(z)$.
Vasco/Admin
PS I have an answer to this exercise which I will post in the very near future on this forum. (Vasco/Admin 13th December 2019).
The version in the 2000 printing contains the following corrected version.
Use the idea behind the method of images to show that if $0<\text{Re}(p)<1$ then
the Green's function of the half-disc $\text{Re}(z)\geq 0,|z|\leq 1$ is
$\displaystyle\mathcal{G}_p(z)=-\ln\bigg|\frac{z-p}{\overline{p}z-1}\bigg|+\ln\bigg|\frac{z+\overline{p}}{pz+1}\bigg|$ .
Check this by getting the computer to draw the level curves of $\mathcal{G}_p(z)$.
Vasco/Admin
PS I have an answer to this exercise which I will post in the very near future on this forum. (Vasco/Admin 13th December 2019).