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