COMPLEX VARIABLES

 

Ma 630

 

Course Description

 

This course extends the concepts of elementary calculus to include the domain of complex numbers.  Topics include: differentiation and integration of complex functions, analytic function, analytic continuation, and Cauchy’s theorems.

 

Goals of the Course

 

1.     To acquaint students with contemporary basic results in the subject.

2.     To explicitly formulate definitions and theorems.

3.     To select exercises that challenge and clarify concepts.

 

Instructional Procedures

 

1.     Lectures/Discussions

2.     Daily homework assignments and in-class discussions of solutions

 

Course Content

 

I.       Complex Numbers

a.      The algebra of complex numbers

b.     Point representation of complex numbers, complex conjugate, and absolute value

c.      Vectors and polar forms

d.     Powers and roots

 

II.    Analytic Functions

a.      Functions of a complex variable

b.     Limits and continuity

c.      Analyticity

d.     The Cauchy-Riemann equations

e.      Harmonic functions

 

III.  Elementary Functions

a.      The exponential function

b.     The logarithmic function

c.      Complex powers

d.     Trigonometric and inverse trigonometric functions

e.      Hyperbolic functions

 

 

IV. Complex Integration

a.      Contours and contour integrals

b.     Independence of path

c.      Cauchy’s integral theorem

d.     Cauchy’s integral formula and its consequences

e.      Bounds for analytic functions

 

V.    Series Representations for Analytic Functions

a.      Sequences and series

b.     Taylor series

c.      Power series

d.     Convergence of series

e.      Laurent series

f.       Zeroes and singularities

g.      The point at infinity

h.      Analytic continuations

 

VI. Residue Theory

a.      The Residue Theorem

b.     Trigonometric integrals over [0, 2 pi]

c.      Improper integrals

d.     Integrals involving multiple-valued functions

e.      The Argument Principle and Rouche’s Theorem

 

VII. Conformal Mapping

a.      Bilinear transformations

b.     The Schwarz-Christoffel Transformation

 

Evaluation Measures

 

1.     Examinations

2.     Problem assignments

3.     Class Participation

 

Bibliography

 

            Required Text:

            Churchill and Brown, Complex Variables and Applications, 5th Ed., McGraw Hill, 1990.

Supporting Bibliography

 

            Ahlfors, Lars, Complex Analysis, 3rd Ed., McGraw Hill, 1979.

 

            Conway, John, Functions of one Complex Variable, 2nd Ed., Springer-Verlag,                           1978.

 

            Fisher, Stephen, Complex Variables, ITP/Brooks Cole Publishing, 1986.

 

            Fuchs, W.H.J., Topics in the Theory of Functions of One Complex Variable, D.                                    Van Nostrand Co., 1967.

 

            Marsden/Hoffman, Basic Complex Analysis, 2nd Ed., W.H. Freeman and Co.,                          1987.

 

            Mathew, John, Complex Variables for Mathematics and Engineers, 2nd Ed., W.C.                                 Brown, 1988.

 

            Spiegel, M., Theory and Problems of Complex Variables, Schaum’s Outline                              Series, McGraw Hill, 1964.

 

            Wunsch, David, Complex Variables with Applications, 2nd ed., Addison-Wesley,                                  1994.