10/15/2015PHY 711 Fall 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

10/15/2015PHY 711 Fall Lecture 222

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10/15/2015PHY 711 Fall Lecture 224

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10/15/2015PHY 711 Fall Lecture 226 Example Re(z) Im(z)

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

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

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

10/15/2015PHY 711 Fall Lecture 2210

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

10/15/2015PHY 711 Fall Lecture 2212 Contour integral for homework:

10/15/2015PHY 711 Fall Lecture 2213 Fourier transforms

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

10/15/2015PHY 711 Fall Lecture 2215

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

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

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10/15/2015PHY 711 Fall Lecture 2220 Example:

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

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

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