Suppose that $z_1 , z_2,\cdots ,z_k,\cdots ,z_d$ are complex variable numbers. Locally, and suppose $f_1, f_2 \cdots, f_k $ are $k$ holomorphic functions on $z_1, z_2 \cdots,z_d$. At $(0,0,...0)\in \mathbb{C}^d$, the Jacobian of $(f_1,\cdots, f_k)$ with respect to $z_1,\cdots, z_k$ is not zero. Rewrite $w_1=f_1, \cdots, w_k=f_k$, can you find another holomorphic functions $w_{k+1},\cdots, w_d$, such that
1) the Jacobian of $w_1,\cdots ,w_d$ with respect to $z_1,\cdots, z_k$ is not zero.
2) for $k<j \leq d$, the partial differential of $w_j$ with respect to $z_1, \cdots ,z_k$ are locally zero.
It seems to me it is a well known result, but I can not find it? Can you give me a reference?