Mathematics & Statistics
Texas Tech University
Kent Pearce
Department of Mathematics and Statistics
Texas Tech University
Lubbock, Texas 79409-1042
Voice: (806)742-2566 x 226
FAX: (806)742-1112
Email: kent.pearce@ttu.edu
Complex Preliminary Examination Topics

  1. Complex Number System and Extended Complex Plane

  2. Properties of Analytic Functions and Harmonic Functions
    1. Cauchy-Riemann Equations
    2. Conformality

  3. Power Series and Analyticity

  4. Properties of Elementary Functions
    1. Mapping Properties
    2. Branch-of-Log
    3. Power Functions
    4. Möbius Transformations

  5. Complex Integration
    1. Complex Line Integral
    2. Index of a Closed Curve
    3. Cauchy's Theorems
    4. Cauchy Integral Formulae
    5. Morera's Theorem
    6. Cauchy-Goursat Theorem

  6. Properties of Analytic Functions
    1. Zeros of an Analytic Function
    2. Identity Theorem
    3. Open Mapping Theorem
    4. Maximum Modulus Theorem
    5. Schwarz's Lemma
    6. Liouville's Theorem
    7. Fundamental Theorem of Algebra

  7. Singularities
    1. Laurent Expansions
    2. Residue Theorem
    3. Argument Principle
    4. Rouché's Theorem
    5. Jensen's Theorem

  8. Spaces of Analytic Functions
    1. Normality and Montel's Theorem
    2. Hurwitz's Theorem
    3. Riemann Mapping Theorem
    4. Weierstrass Factorization

  9. Spaces of Meremorphic Functions
    1. Runge's Theorem
    2. Mittag-Leffler Theorem

  10. Special Functions
    1. Gamma Function
    2. Beta Function
    3. Zeta Function

  11. Harmonic Functions
    1. Dirichlet Problem
    2. Mean-Value Theorem
    3. Maximum Principle
    4. Poisson Formula
    5. Harnack's Theorem
    6. Green's Function

  12. Analytic Continuation
    1. Schwarz Reflection Principle
    2. Schwarz-Christoffel Formula




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Last modified on: Friday, 07-Aug-2015 23:23:09 CDT