# 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.