Course Code: MTH302

Synopsis

Some deep theorems of complex analysis are developed and applications to areas such as fluid mechanics and complex sets are studied. MTH302 follows on from MTH301.
Level: 3
Credit Units: 5
Presentation Pattern: EVERY JULY

Topics

  • Residue theorem.
  • Improper integrals.
  • Modulus of a differentiable function.
  • Schwartz’s lemma.
  • The argument principle.
  • Rouche’s theorem, local mapping and the logarithmic function.
  • Evaluation of real integrals.
  • The probability integral.
  • Analytic continuation.
  • Riemann mapping theorem and Möbious transformations.
  • Theorem on harmonic functions, Julia and Mandelbrot sets.
  • Flows and streamlines.

Learning Outcome

  • Show how to prove a mathematical statement in complex analysis.
  • Calculate the order of zeros and poles of a meromorphic function in a region by the Argument Principle or Rouche's Theorem.
  • Determine suitable linear fractional transformation mapping a region onto another region or the image of a region under a linear fractional transformation.
  • Apply maximum modulus/maximum principle for analytic/harmonic functions.
  • Compute certain improper integrals or the harmonic conjugate of a harmonic function.
  • Demonstrate mathematical reasoning by providing proofs to mathematical statements in complex analysis.
Part-time Programmes / CET Courses