St. Lawrence University

Mathematics, Computer Science & Statistics

Jeremy Ouellette
Asvisor:  Dr. Jim DeFranza
"Physical Applications of Conformal Mapping"

1 Contents 5

2 Properties of Basic Functions 9

2.1 The Complex Exponential…………………………………… 9

2.2 The Complex Logarithm..…………………....……………… 11

2.3 Complex Powers…………………………………………….. 13

2.4 The Bilinear Transformation………………………………… 14

2.5 Schwarz-Christoffel Transformations……………………….. 19

3 Conformal Mappings 23

3.1 Defining a Conformal Mapping……………………………… 23

3.2 Solving Boundary Value Problems…………………………... 25

3.3 Invariance of Laplace’s Equation…………………………….. 26

3.4 The Riemann Mapping Theorem………………………………  27

4 Applications 29

4.1 Steady State Temperatures and Heat Flow…………………… 29

4.2 Electrostatic Potential………………………………………… 34

4.3 Fluid Flow in Two Dimensions………………………………. 39

4.4 Parallel Plate Capacitor……………………………………...... 44

4.5 Other Applications……………………………………………. 47

Created: May 11, 2004