Branch Point/Examples/Cube Root of z-a

Example of Branch Point
Let $f: \C \to \C$ be the complex function defined as:
 * $\forall z \in \C: \map f z = \paren {z - a}^{1/3}$

for some $a \in \C$.

Then $a$ is a branch point of $f$.

Proof
We have that:
 * $f$ has exactly one value at $a$ itself:
 * $\map f a = \paren {a - a}^{1/3} = 0$

and from Cube Roots of Unity we have:


 * $\map f {\map \epsilon 1 + a} = \paren {\paren {\map \epsilon 1 + a} - a}^{1/3} = \map \epsilon 1$
 * $\map f {\map \epsilon 1 + a} = \paren {\paren {\map \epsilon 1 + a} - a}^{1/3} = \map \epsilon {-\dfrac 1 2 + \dfrac {i \sqrt 3} 2 }$
 * $\map f {\map \epsilon 1 + a} = \paren {\paren {\map \epsilon 1 + a} - a}^{1/3} = \map \epsilon {-\dfrac 1 2 - \dfrac {i \sqrt 3} 2 }$

Hence $f$ has more than one value (three in this example) at one or more points in every neighborhood of $a$.