Search the corpus

    Entire functions with $f(z)/z^{n}\to 0$ at infinity are polynomials of degree at most $n-1$

    E-PLJCK

    Area
    Complex Analysis
    Topics
    Cauchy Estimates, Entire Functions, Liouville's Theorem, Polynomials
    Status
    draft

    Suppose \(f\) is entire and suppose that for some integer \(n\geq 1\), \[ \lim_{z\to \infty} {f(z) \over z^n} = 0 .\]

    Prove that \(f\) is a polynomial of degree at most \(n-1\).

    Solution

    Choose \(\abs{z}\) large enough so that \(\abs{f(z)}/\abs{z}^n < \eps\). Then write \(f(z) = \sum_{k\geq 0} c_k z^k\) and estimate \[ 2\pi \abs{c_k} &\leq \oint_{\abs{z} = R} {f(\xi) \over \xi^{k+1}}\dxi \\ &\leq \oint_{\abs{z} = R} {\eps \abs{\xi}^n \over \abs{\xi}^{k+1} } \dxi \\ &= \eps R^{n-(k+1)} \cdot 2\pi R \\ &= \eps C R^{n-k} \\ &\convergesto{\eps \to 0} 0 \] provided \(n-k \leq 0 \iff k\geq n\), since \(\eps \to 0\) forces \(R\to \infty\).

    Source Collections