Example Express each function in the form $w = f(z) = u + iv$, where $u$ and $v$ are real.

$$\begin{array}{ll} 1. & w = z^3 \\ 2. & z = \frac{z}{z+1} \end{array}$$

Answer 1. Let $z = x + iy, w = u + iv$. Then

$$u + iv = z^3 = (x+iy)^3 = x^3 - 3xy^2 + i(3x^2y - y^3)$$

Thus， $u = x^3 - 3xy^2, v = 3x^2y - y^3$.

2. Let $z = x + iy$，$w = u+ iv$. Then

$$u + iv = \frac{z}{z+1} = \frac{z(\bar{z} + 1)}{\vert z + 1\vert^2} = \frac{\vert z\vert^2 + z}{\vert z+1\vert^2}$$

Note that $\vert z\vert^2 = x^2 + y^2$.

$$u + iv = \frac{x^2 + y^2 + x + iy}{(x+1)^2 + y^2}$$

Thus， $u = \frac{x^2 + y^2 + x}{(x+1)^2 + y^2}, \ v = \frac{y}{(x+1)^2 + y^2}$．