Compute the Jacobian matrix of $f(z,w)=(ze^{i\alpha},\ w)$ I have a question:

Let $f$ be a holomorphic and $f(z,w)=(ze^{i \alpha},\ w)$ with $z,\ w \in \Bbb C$. Compute its Jacobian matrix.

I remember that:
For a holomorphic function $f$, its Jacobian matrix is 
$$\left(\begin{array}{cc}\mathrm{Re}~f' & -\mathrm{Im}~f'\\\mathrm{Im}~f' &\mathrm{Re}~f' \end{array}\right)$$
Is is correct? And How do we calculate the Jacobian matrix of $f$?

Any help will be appreciated! Thanks!

Just compute it directly:
$D f((z,w)) = \begin{bmatrix} { \partial f_1(z,w) \over \partial z} &
{ \partial f_1(z,w) \over \partial w} \\
{ \partial f_2(z,w) \over \partial z} &
{ \partial f_2(z,w) \over \partial w}
\end{bmatrix}  = 
\begin{bmatrix} e^{i \alpha} &
0 \\
0 &
1
\end{bmatrix}$
