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D. R. Wilton ECE Dept.

ECE 6382. Functions of a Complex Variable as Mappings. D. R. Wilton ECE Dept. 8/24/10. A Function of a Complex Variable as a Mapping. Simple Mappings: Translations. Simple Mappings: Rotations. Simple Mappings: Dilations. Simple Mappings: Inversions. Inversion – Line-to-Circle Property.

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D. R. Wilton ECE Dept.

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  1. ECE 6382 Functions of a Complex Variable as Mappings D. R. Wilton ECE Dept. 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 For an interpretation of mobius transformations as projections on a sphere, see http://www.youtube.com/watch?v=JX3VmDgiFnY

  13. The Squaring Transformation

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

  15. bottom sheet top sheet The Square Root Transformation

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

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

  18. Constant u and v Contours are Orthogonal

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

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

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

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

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