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Analytical Techniques in Complex Variable Functions

Dive into Laplace transforms, contour integrals, and Cauchy-Riemann equations in this lecture exploring complex variables and their derivatives. Learn about the Cauchy integral theorem and Kramers-Kronig relationships.

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Analytical Techniques in Complex Variable Functions

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  1. PHY 711 Classical Mechanics and Mathematical Methods • 10-10:50 AM MWF Olin 103 • Plan for Lecture 21: • Read Chapter 7 & Appendices A-D • Generalization of the one dimensional wave equation  various mathematical problems and techniques including: • Laplace transforms • Complex variables • Contour integrals PHY 711 Fall 2018 -- Lecture 21

  2. PHY 711 Fall 2018 -- Lecture 21

  3. PHY 711 Fall 2018 -- Lecture 21

  4. PHY 711 Fall 2018 -- Lecture 21

  5. PHY 711 Fall 2018 -- Lecture 21

  6. PHY 711 Fall 2018 -- Lecture 21

  7. PHY 711 Fall 2018 -- Lecture 21

  8. Complex numbers Functions of complex variables Derivatives: Cauchy-Riemann equations PHY 711 Fall 2018 -- Lecture 21

  9. Analytic function PHY 711 Fall 2018 -- Lecture 21

  10. Some details PHY 711 Fall 2018 -- Lecture 21

  11. PHY 711 Fall 2018 -- Lecture 21

  12. PHY 711 Fall 2018 -- Lecture 21

  13. PHY 711 Fall 2018 -- Lecture 21

  14. Example: Im(z) Re(z) PHY 711 Fall 2018 -- Lecture 21

  15. Another example: Im(z) Re(z) PHY 711 Fall 2018 -- Lecture 21

  16. PHY 711 Fall 2018 -- Lecture 21

  17. Cauchy integral theorem for analytic function f(z): PHY 711 Fall 2018 -- Lecture 21

  18. Example Im(z) Re(z) PHY 711 Fall 2018 -- Lecture 21

  19. Example -- continued Im(z) Re(z) PHY 711 Fall 2018 -- Lecture 21

  20. Example -- continued x Im(z’) Re(z’) PHY 711 Fall 2018 -- Lecture 21

  21. Example -- continued Kramers-Kronig relationships PHY 711 Fall 2018 -- Lecture 21

  22. Comment on evaluating principal parts integrals y’ x’ PHY 711 Fall 2018 -- Lecture 21

  23. b(x’) -2L -L x’ 2L L PHY 711 Fall 2018 -- Lecture 21

  24. For our example: b(x) a(x) PHY 711 Fall 2018 -- Lecture 21

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