Let f(z) = log abs(z) + i arg(z) where as usual abs(z) = sqr(x^2+y^2) and arg(z) = tan^(-1)(y/x). (i) Compute, carefully, the Jacobian matrix M.?????...

Let f(z) = log abs(z) + i arg(z) where as usual abs(z) = sqr(x^2+y^2)and arg(z) = tan^(-1)(y/x).(i) Compute, carefully, the Jacobian matrix M.?????(ii) You should be able to factor a term 1/abs(z) out of every entry in the matrix, and view what is left as a rotation by some angle theta?????(iii) Multiplication by z is a dilation by abs(z) and rotation by arg(z). Multiplication by M does not correspond to multiplying by z, but rather what complex number????(iv) You have just proven a formula for f(z); what is it????