So I have the book Analytic Function Theory Vol. II by Einar Hille and have been working through the exercises as I go.
Now, let's take a look at Exercise 14.4 question 8 (since this ties into question 9):
Show that $\log(1+|w|^2)^{\frac12}$ satisfies the equation$$\Delta F=\dfrac{\partial^2F}{\partial u^2}+\dfrac{\partial^2F}{\partial v^2}=2e^{-4F},w=u+iv$$
Now this was pretty easy to do (once I realized that $\log$ was supposed to represent $\ln$ for some reason):
We have$$\ln(1+|w|^2)^{1/2}=\dfrac12\ln(1+u^2+v^2)$$And then we can take $\dfrac{\partial F}{\partial u}$ to be$$\dfrac u{u^2+v^2+1}$$and then take $\dfrac{\partial^2F}{\partial u^2}$ to be$$\dfrac{v^2-u^2+1}{(u^2+v^2+1)^2}$$Then taking $\dfrac{\partial F}{\partial v}$ and $\dfrac{\partial^2F}{\partial v^2}$ is trivial, and we get$$\dfrac{\partial^2F}{\partial u^2}+\dfrac{\partial^2F}{\partial v^2}=\dfrac2{(u^2+v^2+1)^2}$$and from there it is easy to verify that$$\dfrac{\partial^2F}{\partial u^2}+\dfrac{\partial^2F}{\partial v^2}=2e^{-4F}$$
And now, here's question 9:
Use the preceding result to evaluate directly the integral (14.4.18). Take $g(z)=\log[1+|f(z)|^2]^{\frac12}$ and apply Green's theorem,$$\iint_D\Delta GdS=\int_{\partial D}\dfrac{\partial G}{\partial n}ds$$where $D=D_\delta$ is the disk $|z|\lt t$ from which disks of radius $\delta$ about each of the poles of $f(z)$ have been deleted. Let $\delta\to0$. Differentiation is taken in the direction of the outer normal in the right member.
Note that (14.4.18) references this integral:$$\pi A(t;f)=\int_{|z|\lt t}\dfrac{|f'(z)|^2d\omega}{[1+|f(z)|^2]^2}$$however, my question is: What would I do to attain the solution for this problem? I don't exactly know where I should start when doing a problem like this.
If anyone is confused about what chapter this is in, this is "Entire and Meromorphic functions".