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    Sameer Chavan — Problems in Complex Analysis

    SRC-CHAVAN-COMPLEX-PROBLEMS-2015

    Area
    Complex Analysis
    Status
    draft
    Source
    Contents

    25 problems.

    A Maximum Modulus Principle for Analytic Polynomials

    1. Maximum modulus for polynomials via a unitary dilation
    2. Maximum modulus for polynomials from circular mean squares
    3. Maximum modulus for a uniformly convergent power series on the closed disk

    Zeros of Analytic Polynomials

    1. Fundamental theorem of algebra from a minimum-modulus estimate
    2. A Cauchy-Schwarz bound on the zeros of a monic polynomial

    Argument Principle and its Consequences

    1. Discontinuities of the principal argument on the punctured plane
    2. An analytic square root without an analytic logarithm
    3. A boundary-modulus-one holomorphic map covers the unit disk
    4. A unique real solution of \(\lambda=z+e^{-z}\) in the right half-plane
    5. Count zeros of \(3e^z-z\) in the closed unit disk

    Hurwitz’s Theorem

    1. A partial sum of the cosine series has a zero near the origin
    2. A compact limit of injective holomorphic functions is constant or injective

    Open Mapping Theorem

    1. Moduli of an open subset of the complex plane form a relatively open set
    2. Maximum modulus principle for open mappings
    3. The boundary of the image of a relatively compact domain lies in the image of its boundary
    4. Minimum modulus principle

    Schwarz’s Lemma

    1. The automorphism group of the unit disk acts transitively

    Simple Connectivity and Cauchy’s Theorem

    1. The complex plane minus a half-line is simply connected
    2. The complex plane minus a finite nonempty set is not simply connected

    Range of a Holomorphic Function

    1. The range of a nonconstant entire function is dense
    2. No nonconstant entire function omits a half-line
    3. Holomorphic automorphisms of \(\CC\) are the maps \(z\mapsto az+b\) with \(a\neq 0\) — Problem 8.7; holomorphic automorphisms of C are exactly z -> az+b with a nonzero

    Zeros of Analytic Polynomials in Several Variables

    1. Invertible complex matrices are dense
    2. Zeros of a multivariable complex polynomial are not isolated
    3. The complex general linear group is path-connected

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