1. D. R. Wilton ECE Dept. ECE 6382 Functions of a Complex Variable as Mappings 8/24/10

2. A Function of a Complex Variable as a Mapping

3. Simple Mappings: Translations

4. Simple Mappings: Rotations

5. Simple Mappings: Dilations

6. Simple Mappings: Inversions

7. Inversion – Line-to-Circle Property

8. Inversion – Line-to-Circle Property, cont’d

9. Simple Mappings: Inversions, cont’d

10. A General Linear Transformation (Mapping) Is a Combination of Translation, Rotation, and Dilation

11. A General Bilinear Transformation (Mapping) Is a Succession of Translations, Rotations, Dilations, and Inversions

12. Bilinear Transformation Example: The Smith Chart Bilinear Transformation Example: The Smith Chart For an interpretation of mobius transformations as projections on a sphere, see http://www.youtube.com/watch?v=JX3VmDgiFnY

13. The Squaring Transformation

14. Another Mapping of the Squaring Transformation 1 23 3 2 1 9 4 1 90 o 180 o 270 o 360 o -180 o -270 o -360 o 0o0o -90 o Re Im

15. The Square Root Transformation top sheet bottom sheet

16. Connecting the Two Sheets at a Branch Cut to Form a Riemann Surface bottom sheet branch point branch cut Only at a branch point are multiple cycles encircling the point required to return to the starting value top sheet

17. The Branch Cut Allows Us to Map the Square Root Function 1 23 3 2 1 1 22.5 o 45 o 67.5 o 90 o -45 o -67.5 o -90 o 0o0o -22.5 o top sheet, k = 0 1 23 3 2 1 1 202.5 o 225 o 247.5 o 270 o 135 o 90 o 180 o 157.5 o bottom sheet, k = 1 112.5 o Re Im

18. Constant u and v Contours are Orthogonal

19. Mappings of Analytic Functions are Conformal (Angle-Preserving)

20. Constant |w| and Arg w Contours are also Orthogonal -45 o 1 23 3 2 1 1 22.5 o 45 o 67.5 o 90 o -67.5 o -90 o 0o0o -22.5 o

21. The Logarithm Function Infinite # of sheets Branch cut Branch point Riemann surface for the log function

22. Arbitrary Powers of Complex Numbers Infinite # of sheets Branch cut Branch point Riemann surfaces for z a