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    An analytic square root without an analytic logarithm

    P-CHAVAN-CA-3-3

    Area
    Complex Analysis
    Status
    draft
    Contents

    Let \[ \Omega=\mathbb C\setminus[-1,1], \qquad f(z)=z^2-1, \] and define \[ g(z)=|f(z)|^{1/2} \exp\!\left(\frac{i}{2}\bigl(\arg(z-1)+\arg(z+1)\bigr)\right), \] where \(\arg\) is the principal argument.

    1. Show that \(g\) is well-defined and continuous on \(\Omega\) and satisfies \(g^2=f\).

    2. Show that \(g\) is analytic on \(\Omega\).

    3. Show that \(f\) has no analytic logarithm on \(\Omega\).

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