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18.112

Functions of a Complex Variable

3-0-9 units · Undergraduate

Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of non-Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions.

Fall 2008

Topics

  • The algebra of complex numbers: the geometry of the complex plane, the spherical representation
  • Exponential function and logarithm for a complex argument: the complex exponential and trigonometric functions, dealing with the complex logarithm
  • Analytic functions; rational functions: the role of the Cauchy-Riemann equations
  • Power series: complex power series, uniform convergence
  • First in-class test
  • Exponentials and trigonometric functions
  • Conformal maps; linear transformations: analytic functions and elementary geometric properties, conformality and scalar invariance
  • Linear transformations (cont.): cross ratio, symmetry, role of circles
  • Line integrals: path independence and its equivalence to the existence of a primitive
  • Cauchy-Goursat theorem
  • The special Cauchy formula and applications: removable singularities, the complex Taylor’s theorem with remainder
  • Isolated singularities
  • The local mapping; Schwarz’s lemma and non-Euclidean interpretation: topological features, the maximum modulus theorem
  • The general Cauchy theorem
  • The residue theorem and applications: calculation of residues, argument principle and Rouché’s theorem
  • Contour integration and applications: evaluation of definite integrals, careful handling of the logarithm
  • Harmonic functions: harmonic functions and holomorphic functions, Poisson’s formula, Schwarz’s theorem
  • Second in-class test
  • Mittag-Leffer’s theorem: Laurent series, partial fractions expansions
  • Infinite products: Weierstrass’ canonical products, the gamma function
  • Normal families: equiboundedness for holomorphic functions, Arzela’s theorem
  • The Riemann mapping theorem
  • The prime number theorem: the history of the theorem and the proof, the details of the proof
  • The extension of the zeta function to C, the functional equation

Readings

  • Ahlfors, Lars V. Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable . 3rd ed. New York, NY: McGraw-Hill, 1979. ISBN: 9780070006577.
    Listed reading

Prerequisites & course information

Subject meets with 18.1121 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) U (Fall) 3-0-9 units

Sources