Singapore University of Social Sciences

Applied Complex Analysis (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 January

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.
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