What is the Jacobian? What is the Jacobian of the function $f(u+iv)={u+iv-a\over u+iv-b}$?
I think the Jacobian should be something of the form  $\left(\begin{matrix}
  {\partial f_1\over\partial u} & {\partial f_1\over\partial v}  \\
  {\partial f_2\over\partial u} & {\partial f_2\over\partial v}
 \end{matrix}\right)$
but I don't know what $f_1,f_2$ are in this case. Thank you.

You could just write $(u+iv−(a_1+a_2 i))/(u+iv−(b_1+b_2 i))$ where $u,v,a_1,a_2,b_1,b_2$ are real.  Then multiply the numerator and denominator by the complex conjugate of the denominator to find the real and imaginary parts.
Then later, exploit the Cauchy--Riemann equations to conclude that the matrix must have the form $\begin{bmatrix} {}\ \ \  c & d \\ -d& c\end{bmatrix}$ where $c$ and $d$ are some real numbers and $f\;'(z)=c+id$.
