Kutta-Joukowski theorem applied on a Joukowski airfoil (derivation)

I have a doubt about the derivation of the Kutta-Joukowski theorem for a Joukowski airfoil. I know the results, but my main objective is to know how get these ones.

Consider for the initial plane a cylinder centered on $zeta_0$, with a circulation -$Gamma$, in an uniform flow with an atack angle $alpha$:

where $a$ is the circumference radius, and $b$ the intersection of the circumference with the real positive axis, $xi$. The parameter $beta$ is the angle between the horizontal line and the line that links $zeta_0$ to $b$. The center of the circumference is:

$$zeta_0=b-ae^{-ibeta}$$

For this flow we have the following complex potential:

$$W(zeta)=U_inftyleft[e^{-ialpha}left(zeta-zeta_0right)+frac{e^{ialpha}a^2}{left(zeta-zeta_0right)}right]+frac{iGamma}{2pi}lnleft(zeta-zeta_0right)$$

The Joukowski transform is:

$$z=zeta+frac{b^2}{zeta}$$

To determine the complex potential on the $z$ plane we need to find the relation between $zeta$ and $z$. And we get:

$$zeta=frac{z}{2}pmsqrt{left(frac{z}{2}right)^2-b^2}$$

That is an awful relation. We not only have a square root, but also an $pm$ symbol. Substituting this on the potential complex and then find the residues of $left(frac{dW}{dz}right)^2$ and $left(frac{dW}{dz}right)^2 z$ would be pratically impossible. What should be the next steps?

[Note: I didn’t find any paper or book that shows the derivation of this theorem on the $z$ plane, but only on the $zeta$ plane, which I already know. Do you know anything that can help me?]