MA 234 - Complex Variables with Applications

MA 234 - Complex Variables with Applications

An introduction to functions of a complex variable.  The   topics covered include complex numbers, analytic and harmonic functions, complex integration, Taylor and Laurent series, residue theory, and improper and trigonometric integrals.

Corequisites:    MA 227

Course Objectives

This is an introductory course in complex analysis.  Upon completion, students should have a working knowledge of the basic definitions and theorems of the differential and integral calculus of functions of a complex variable and know the similarities and differences between real and complex analysis.

Learning Outcomes

  1. Complex arithmetic, algebra and geometry:  Develop facility with complex numbers and the geometry of the complex plane culminating in finding the n nth roots of a complex number.
  2. Differentiable Functions and the Cauchy-Riemann equations:  Show knowledge of whether a complex function is differentiable and use the use the Cauchy-Riemann equations to calculate the derivative.
  3. Analytic and Harmonic functions:  Determine if a function is harmonic and find a harmonic conjugate via the Cauchy-Riemann equations.
  4. Sequences, Series and Power Series:  Determine whether a complex series converges.  Show understanding of the region of convergence for power series.
  5. Elementary functions – exponential and logarithm:  Understand the similarities and differences between the real and complex exponential function.  Compute the complex logarithm.
  6. Elementary functions – trigonometric and hyperbolic:  Understand the relationships among the exponential, trigonometric and hyperbolic functions. Derive simple identities. 
  7. Complex integration – contour integrals:  Set up and directly evaluate contour integrals
  8. Complex integration – Cauchy’s Integral Theorem and Cauchy’s Integral Formula:  Identify when the theorems are applicable and evaluate contour integrals using the Cauchy Integral Theorem and the Cauchy Integral Formula in basic and extended form.
  9. Taylor and Laurent Series:  Find Taylor or Laurent Series for simple function.  Show understanding of the convergence regions for each type of series.
  10. Singularities, zeros and poles:  Identify and classify zeros and singular points of functions.
  11. Residue Theory:  Compute residues.  Use residues to evaluate various contour integrals.

Textbooks

Mathers, John H. and Russell W. Howell, Complex Analysis for Mathematics and Engineering, fifth edition, Jones & Bartlett