Linear approximation of a complex function

Question

Consider the function $w = z^3$ near the point $z = 1 + i, w = −2 + 2i$. What is the linear approximation near this point for this mapping? Is the linear approximation a one-to-one function? If so, find its inverse.

Answer 1

The linear approximation of a function $f(z)$ around a point $a$ is $f(a)+f'(a)(z-a)$. Here we have $a=1+i$ and $f'(z)=3z^2$ which means $f'(a)=3(1+i)^2=6i$. This means our linear approximation is $$w=-2+2i+6i(z-(1+i)).$$

Because this function is linear it must be one-to-one. The inverse is $$\frac{w-4+4i}{6i}=z.$$