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Thread: Starlike Set

  1. #1
    Senior Member slevvio's Avatar
    Oct 2007

    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
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  2. #2
    Senior Member Tinyboss's Avatar
    Jul 2008
    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.
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