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    A conformal equivalence from the quarter-disc to the first quadrant

    E-4EZMD

    Area
    Complex Analysis
    Topics
    Biholomorphisms, Conformal Maps
    Status
    draft

    Define \(A \da \ts{\Re(z) > 0, \Im(z) > 0}\). Find a conformal equivalence \(\Delta \intersect A \to A\).

    Solution

    The composite of the following three maps is a conformal equivalence \(\Delta\intersect A\to A\).

    • The squaring map \(z\mapsto z^2\) sends \(\Delta\intersect A\) onto \(\DD \intersect \HH\).

    • The negated Joukowski map \(z\mapsto -{1\over 2}(z+\inverseof{z})\) sends \(\DD\intersect\HH\) onto \(\HH\).

    • The principal branch of \(z\mapsto z^{1\over 2}\) sends \(\HH\) onto the first quadrant \(A\).

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