# Thread: Starlike Set

1. ## Starlike Set

Is the set

$\displaystyle C := \mathbb{C} \setminus \{x \in R : x \le -1\}$

a starlike set with star-centre 0?

I ask this because I need the result

$\displaystyle \frac{d}{d z} \left[ \int_{[0,z]} \frac{d \xi}{1 + \xi} \right] = \frac{1}{1+z}$, where [0,z] means the line segment from the star centre 0 to any point $\displaystyle z \text{ in } C$. Here f: $\displaystyle C \rightarrow \mathbb{C}$ is continuous and analytic on $\displaystyle C$ and so its integral around any closed path in C = 0, but only if C is starlike. So the integral around any triangular path in particular is equal to 0, which tels us we can differentiate the integral on C.

However I am not sure about C being starlike! Any help would be appreciated

2. The reason the theorem requires the set to be "starlike with star-center 0" is precisely so that the line [0,z] will be contained in the set (since you're integrating over that path). Just check whether [0,z] is in your set for every point z in the set, which is the definition of starlike with star-center 0.