Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Interactive Visualization in Complex Analysis

Matthias Kawski

Dept. of Math. & Statistics

Arizona State University

Tempe, Arizona U.S.A.

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Thanks for generous support by

Arizona State University: College of Liberal Arts and Sciences:

“Interactive Visualization across the Mathematics Curriculum”

INTEL Corporation through grant 98-34

“Interactive visualization“

National Science Foundation through the grants

DUE 97-52453 “Vector Calculus via Linearization:

Visualization and Modern Applications”

DMS 00-72369 “Geometry and algebra of nonlinear control systems”

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Outline

Introduction. Objectives

Examples

Mapping curves – too easy? Do it right!

Mapping domains (MAPLE)

Color coding, zoom essential singularity (JAVA)

Convergence (not yet fully interactive, MATLAB)

Conclusion

Experiences

What’s next? Literature.

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Some goals

Complex Analysis

“analytic” functions: a very strong hypothesis,

that guarantees amazingly rich properties

beauty, symmetry, “harmony” (-nic) wherever one looks

Mathematics

“doing mathematics”: (guided) exploration observation conjecture

testing formulate theorem attempt proof

polish proof/thm w/ making definitions axiomatize

enjoy the beauty

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Math in the 3rd millennium A.D.

Utilize a richer symbolic language than just single Latin and Hebrew characters…. … instead manipulate “iconified” objects (J.Mason), right-click on “objects” to infer their properties

“interactive” exploration “visual language” “tangible” – click, drag, and draw

want students to learn to “do” mathematics, … not just recite old results (which are archived and everywhere always accessible electronically ….)

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

So many great examples…..

Selected references – almost all on-line. See conference paper (on-line, too!!!) for details Abdo, G., Godfrey, P., “Plotting Functions of a Complex Variable”, Florida Institute of Technology

Akers, D., ”g(z): A Tool For Visual Complex Analysis”, Brown University

Arnold, D., “Graphics for Complex Analysis”, Pennsylvania State University

Banchoff, T., Cervone, D.,“Understanding Complex Function Graphs”, Brown University and The Geometry Center also: Communications in Visual Mathematics, vol.1, no.1, (1998).

Bennett , A., “Complex Function Grapher “, Journal of Online Mathematics and Applications

Frank Farris, “Complex Function Visualization”, Santa Clara University

Fishback, P., “Resources for the Teaching of Complex, Grand Valley State University

Joyce, D., ”Julia and Mandelbrot Set Explorer”, Clark University

Kawski, M., commented MAPLE-worksheet directory for complex analysis

Kawski, M., JAVA and Complex Analysis

“f(z) - The Complex Variables Program, Lascaux Graphics

Lundmark , H., “Visualizing complex analytic functions using domain coloring, Linköping University

Needham, T., “Visual Complex Analysis”, 1998, Oxford University Press

Orpen, K., and Djun, M., “Java Complex Function Viewer”, University of British Columbia

Santa Cruz, S., and Soares Fonseca, P., "BOMBELLI -A Java viewer for arbitrary, user-specified complex functions , Universidade Federal de Pernambuco, Brazil.

Websites related to "Visual Complex Analysis"

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Mapping curves too easy? “Dots” are critical! Inversion z -> 1/z as a special case of a Moebius trafo

The visual impact of dragging / moving the object is a compelling connection to physics and E-statics

Construct from basic geometric

principles, play and explore,

observe and formalize (here:

nonuniform parameterization),

then rigorously establish general

properties of conformal mappings

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Mapping domains Use color- coded grid!

Very hard to visualize w/o tools

Almost ideal for study via

interactive visualization

Let’s do it: Go to MAPLE….

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Complex functions color-coded

Color coding adapted from Frank Farris at Santa Clara U, see also Needham’s Visual Complex Analysis WWW-site

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Complex singularities

Let’s do it!

Go to JAVA!

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Convergence of Laurent series

Uniform convergence

on compact subsets

of open annulus !!?

Why the asymmetry?

“the approximant interpolates

the original function at

asymptotically uniformly

spaced points on the circles

of convergence”. Proof?

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Uniform convergence of ((the error term of))

a Laurent series on compact subsets of annulus

Mesmerizing beauty!

Observations, Questions, Conjectures, Proofs, …

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Winding number & branch-cuts What are the exercises,

questions, desired insights

and discoveries?

This is just

a simple JAVA

proof-of-concept,

try it out – now ! plan for full “parser

(inverse fcns, branch-cuts!)

Suitable for many honors

visualization projects

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Classroom experience

Caveat:

Only one small and very mixed class in 2000

Just a few minutes of play at the beginning provide

material for class and homework to analyze

Students “demand” proof! (as opposed to:

“why do we have to prove it if it is not going on the test anyhow?”)

Students’ teaching evaluations

Enrollment in subsequent semesters (same, and others by recommendation)

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Where to find more?

Just search the net!

Math-archives

Needham: “Visual Complex Analysis”

See reference in this conference article

All my papers, applets, worksheets, and most talks

are available on-line. Free, of course.

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization Complex Analysis 2nd ICTM Hersonissos, Crete July 2002

http://math.asu.edu/~kawski kawski@asu.edu