Does there exist a function $f : \mathbb{C}^n \to (\mathbb{R}^+)^{n}$ that involves the phase of its parameters and the sum of its result is $1$?

Question

I am interested in finding a function $f: \mathbb{C}^{n} \to (\mathbb{R}^+)^{n}$ with properties as follows.

Let $\mathbf{u} = \left[u_1\ u_2\ u_3\ \dots\ u_n \right]^T$ and $\mathbf{v} = \left[v_1\ v_2\ v_3\ \dots\ v_n \right]^T = f(\mathbf{u})$. The value of $v_1 + v_2 + \dots + v_n$ must be equal to $1$.

However, a function that satisfied condition above is a function $f(\mathbf{z})$ such that $$v_k = \frac{e^{|z_k|}}{e^{|z_1|} + e^{|z_2|} + \dots + e^{|z_n|}}$$

But this function does not involve the phase of its parameters. Only its magnitude.

Does anyone have an example of the function that involves all of the magnitude and phase of its parameters?