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    The number of solutions of $6z^3+1=-e^z$ in $|z|<1$

    P-URPZ4

    Area
    Complex Analysis
    Topics
    Rouché, Zeros
    Status
    draft

    Find the number of solutions to the following equation on \(\abs{z} < 1\): \[ 6z^3 + 1 = -e^z .\]

    Solution

    Write \(f(z) \da 6z^3 + 1 + e^z\).

    • Small: \(m(z) = e^z + 1\)
    • Large: \(M(z) = 6z^3\)
    • The estimate: \[ \abs{m(z)} = \abs{e^z + 1} \leq e^{\Re(z)} + 1 \leq e^{\abs{z}} + 1 = e+1 < 6 = \abs{6z^3} ,\] so \(3 = Z_M = Z_f\).

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