1. 10/15/2015PHY 711 Fall 2014 -- Lecture 221 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 22: Summary of some mathematical tools 1.Contour integration 2.Fourier transforms 3.Fast Fourier transforms

2. 10/15/2015PHY 711 Fall 2014 -- Lecture 222

3. 10/15/2015PHY 711 Fall 2014 -- Lecture 223

4. 10/15/2015PHY 711 Fall 2014 -- Lecture 224

5. 10/15/2015PHY 711 Fall 2014 -- Lecture 225

6. 10/15/2015PHY 711 Fall 2014 -- Lecture 226 Example Re(z) Im(z)

7. 10/15/2015PHY 711 Fall 2014 -- Lecture 227 Example -- continued Re(z) Im(z)

8. 10/15/2015PHY 711 Fall 2014 -- Lecture 228 Example -- continued Re(z) Im(z) x’

9. 10/15/2015PHY 711 Fall 2014 -- Lecture 229 Example -- continued Kramers-Kronig relationship

10. 10/15/2015PHY 711 Fall 2014 -- Lecture 2210

11. 10/15/2015PHY 711 Fall 2014 -- Lecture 2211 Example: Re(z) Im(z)

12. 10/15/2015PHY 711 Fall 2014 -- Lecture 2212 Contour integral for homework:

13. 10/15/2015PHY 711 Fall 2014 -- Lecture 2213 Fourier transforms

14. 10/15/2015PHY 711 Fall 2014 -- Lecture 2214 Note: The location of the 2  factor varies among texts.

15. 10/15/2015PHY 711 Fall 2014 -- Lecture 2215

16. 10/15/2015PHY 711 Fall 2014 -- Lecture 2216 Use of Fourier transforms to solve wave equation

17. 10/15/2015PHY 711 Fall 2014 -- Lecture 2217 Use of Fourier transforms to solve wave equation -- continued

18. 10/15/2015PHY 711 Fall 2014 -- Lecture 2218

19. 10/15/2015PHY 711 Fall 2014 -- Lecture 2219

20. 10/15/2015PHY 711 Fall 2014 -- Lecture 2220 Example:

21. 10/15/2015PHY 711 Fall 2014 -- Lecture 2221 Example: f(t) F(  )

22. 10/15/2015PHY 711 Fall 2014 -- Lecture 2222 F(  ) =-M =M

23. 10/15/2015PHY 711 Fall 2014 -- Lecture 2223

24. 10/15/2015PHY 711 Fall 2014 -- Lecture 2224

25. 10/15/2015PHY 711 Fall 2014 -- Lecture 2225

26. 10/15/2015PHY 711 Fall 2014 -- Lecture 2226 Cooley-Tukey algorithm: J. W. Cooley and J. W. Tukey, “An algorithm for machine calculation of complex Fourier series” Math. Computation 19, 297-301 (1965)