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    Entire functions with $f(z)\to\infty$ as $z\to\infty$ are polynomials

    E-SUDRI

    Area
    Complex Analysis
    Topics
    Casorati-Weierstrass, Entire Functions, Polynomials, Singularities
    Status
    draft

    Show that if \(f\) is entire and \(f(z) \convergesto{z\to\infty} \infty\) then \(f\) is a polynomial.

    Solution

    • Set \(g(z) \da f(1/z)\), so \(g(z) \convergesto{z\to 0} \infty\) making \(z=0\) a singularity.

    • This is not an essential singularity by Casorati-Weierstrass.

    • So this is a pole and \(g(z) = \sum_{-N\leq k \leq 0} c_k z^k\) for \(N\) the order of the pole

    • Thus \(f(z) = \sum_{0<k<N}c_k z^k\) is a polynomial.

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