# 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

**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.**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.**Analytic and Harmonic functions:**Determine if a function is harmonic and find a harmonic conjugate via the Cauchy-Riemann equations.**Sequences, Series and Power Series:**Determine whether a complex series converges. Show understanding of the region of convergence for power series.**Elementary functions – exponential and logarithm:**Understand the similarities and differences between the real and complex exponential function. Compute the complex logarithm.**Elementary functions****–****trigonometric and hyperbolic:**Understand the relationships among the exponential, trigonometric and hyperbolic functions. Derive simple identities.**Complex integration – contour integrals:**Set up and directly evaluate contour integrals**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.**Taylor and Laurent Series:**Find Taylor or Laurent Series for simple function. Show understanding of the convergence regions for each type of series.**Singularities, zeros and poles:**Identify and classify zeros and singular points of functions.**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