1. ECE 6382 Functions of a Complex Variable as Mappings Notes are adapted from D. R. Wilton, Dept. of ECE 1 Davi R. Jackson Notes 4

2. A Function of a Complex Variable as a Mapping 2

3. Simple Mappings: Translations 3

4. Simple Mappings: Rotations 4

5. Simple Mappings: Dilations 5 Note:

6. A General Linear Transformation (Mapping) is a Combination of Translation, Rotation, and Dilation 6 Shapes do not change under a linear transformation!

7. Simple Mappings: Inversions 7

8. Circle Property of Inversion Mapping 8 Consider a circle: This is in the form Hence (This maps circles into circles.) J. W. Brown and R. V. Churchill, Complex Variables and Applications, 9 th Ed., McGraw-Hill, 2013.

9. Circle Property of Inversion Mapping (cont.) 9 Multiply by u 2 + v 2 : This is in the form of a circle: or

10. Simple Mappings: Inversions (cont.) 10 Shapes are not preserved! Note the circular boundaries for the region!

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

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

13. The Squaring Transformation 13

14. Another Representation 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 14

15. The Square Root Transformation 15  We say that there are two “branches” (i.e., values) of the square root function.  Note that for a given branch (e.g., the principal branch), the square root function is not continuous on the negative real axis. (There is a “branch cut” there.) Principal branch Second branch Note: The value of z 1/2 on one branch is the negative of the value on the other branch.

16. The Square Root Transformation (cont.) 16 Principal branch Second branch The principal square root is denoted as The principal branch is the choice in MATLAB and most programming languages! Note:

17. 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 Principal branch, 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 Other branch, k = 1 112.5 o 17 The Square Root Transformation (cont.)

18. Constant u and v Contours are Orthogonal 18

19. Constant u and v Contours are Orthogonal (cont.) Example: so Also, recall that 19

20. Mappings of Analytic Functions are Conformal (Angle-Preserving) 20 Hence

21. Constant u and v Contours are Orthogonal (Revisited) Since the contours u = constant and v = constant are (obviously) orthogonal in the w plane, they must remain orthogonal in the z plane. 21

22. Constant | w | and arg( w) Contours are also Orthogonal 22

23. The Logarithm Function 23 There are an infinite number of branches (values) for the ln function!

24. Arbitrary Powers of Complex Numbers 24 ( a may be complex)

25. Arbitrary Powers of Complex Numbers (cont.) For z p/q the repetition period is k = q. For irrational powers, the repetition period is infinite; i.e., values never repeat! 25