I.  Number and Title of Course

  MAT 411  Introduction to Complex Variables
 

II. Reasons for Revision

We have updated the bibliography to better address and reflect the current work in this area, and we have redefined the major objectives of the course in terms of student outcomes.  The course continues to serve the following purposes in our program:

A. To present basic concepts in the theory of functions of a complex variable not covered in any detail in the other courses we currently offer.

B. An introduction to the theory of functions of a complex variable is an essential part of mathematical analysis, valuable for anyone interested in pure or applied mathematics.
 

III. Major Objectives of the Course

Students will:

A. Come to understand the complex number system as an extension of the reals.

B. Investigate properties of elementary complex functions.

C. Learn the theory of functions of a complex variable through mappings, differential and integral calculus, power series, and residues.
 

IV. Topical Outline

A. Complex Numbers

 1.  Definition, sums and products
 2.  Algebraic properties and geometric representation
 3.  Moduli and conjugates
 4.  Polar coordinates and exponential form
 5.  Roots of complex numbers
 6.  Regions in the complex plane

B. Analytic Functions

 1.  Functions of a complex variable
 2.  Limits and continuity
 3.  Derivatives
 4.  Differentiation formulas
 5.  Cauchy-Riemann equations
 6.  Analytic functions
 7.  Harmonic functions

 1.  The exponential function
 2.  Trigonometric functions
 3.  Hyperbolic functions
 4.  The logarithm function
 5.  Properties of these functions
 6.  Complex exponents
 7.  Inverse trigonometric and hyperbolic functions

D. Integrals

 1.  Complex valued functions
 2.  Contours and contour integrals
 3.  Cauchy-Goursat theorem
 4.  Simply and multiply connected domains
 5.  Cauchy integral formula
 6.  Liouville's and Morera's theorems

E. Series

 1.  Taylor series
 2.  Laurent series
 3.  Absolute and uniform convergence of series
 4.  Integration and differentiation of series
 5.  Multiplication and division of series

F. Residues and Poles

 1.  Residues
 2.  Residue theorem
 3.  Residues at poles
 4.  Evaluation of improper integrals
 5.  Integration along a branch cut

G. Mapping

 1.  Linear transformations
 2.  Linear fractional transformations
 3.  Exponential and logarthmic transformations
 4.  The transformation w = sin z , w = sinh z , w = cos z

H. Conformal Mapping

 1.  Preservation of angles
 2.  Harmonic conjugates
 3.  Transformations of harmonic functions
 4.  Applications of conformal mappings to steady state temperatures

V.   Bibliography

Ahlfors, L. V. Complex Analysis, 3rd ed., McGraw-Hill, Inc., New York, 1979.
Bieberbach, L. Conformal Mapping, Chelsea Publishing Co., New York, 1986.
Boas, R.P., Invitation to Complex Analysis, McGraw-Hill, Inc., New York, 1987.
Brown, J.W., and R.V. Churchill, Fourier Series and Boundary Value Problems, 5th ed., McGraw-Hill, Inc., New York, 1993.
Brown, J.W., and R.V. Churchill, Complex Variables and Applications, 6th ed.,McGraw-Hill, Inc., New York, 1996.
Dettman, J.W. Applied Complex Variables, Dover Publications, Inc., New York, 1984.
Fisher, S.D. Complex Variables, Wadsworth, Inc. Belmont, CA, 1986.
Henrici, P. Applied and Computational Complex Analysis, Vols. 1, 2, and 3, John Wiley & Sons, Inc., 1988, 1991, and 1993.
Kaplan, W. Advanced Calculus, 4th ed., Addison-Wesley Publishing Co. Inc., Reading, MA, 1991.
Krantz, S.G. Complex Analysis: the Geometric Viewpoint, Carus Mathematical Monagraph Series, The Mathematical Association of America, Washington, DC,1990.
Lang, S. Complex Analysis, 3rd ed., Springer-Verlag, New York, 1993.
Levinson, N. and R.M. Redheffer Complex Variables, McGraw-Hill, Inc., New York, 1988.
Marsden, J.E. and M.J. Hoffman Basic Complex Analysis, 2nd ed., W. H. Freeman and Company, New York, 1987.
Mathews, J.H. Complex Variables for Mathematics and Engineering, 2nd ed., Wm. C. Brown Publishers, Dubuque, IA, 1988.
Palka, B.P. An Introduction to Complex Function Theory, Springer-Verlag, New York, 1990.
Rubenfeld, L.A. A First Course in Applied Complex Variables, John Wiley & Sons, Inc., New York, 1985.
Saff, E.B. and A.D. Snider Fundamentals of Complex Analysis for Mathematics Science, and Engineering, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, NJ, 1993.
 

VI.  Presentation and Evaluation

A. Presentation will include lectures, and class discussions.

B. Evaluation will be based upon problem assignments and written examinations.
 

VII. Prerequisite

  MAT 263 (MAT 270 is recommended)
 

VIII. Credit

  3 credits:  (3.0)
 

IX.  Departmental Approval

This course proposal was examined in accordance with recommended procedures and
approved by the Department of Mathematics Curriculum Committee on
_____________________.    _____________________________________
             Signature of Department Chair   Date
 

X.   Catalog Description

Complex numbers, analytic functions, elementary functions, contour integration, integral theorems, Taylor series, Laurent series, uniform convergence, calculus of residues, mappings and applications.
 XI.  Qualifications of Faculty Who Will Teach the Course

A. An M.S. or M.A. degree in Mathematics or a related field and an interest and training in complex variables.

B. Some of the current faculty who may teach the course:
 James Guyker, Ph.D. in Mathematics
 Peter Mercer, Ph.D. in Mathematics
 

XII. Support Services Required

 Present classroom facilities are adequate.