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Blue book

Fourier Methods. Fourier sine series Application to the wave equation Fourier cosine series Fourier full range series Complex form of Fourier series Introduction to Fourier transforms and the convolution theorem. Blue book. New chapter 12. Dr Mervyn Roy (S6 ). Last time.

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Blue book

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  1. Fourier Methods • Fourier sine series • Application to the wave equation • Fourier cosine series • Fourier full range series • Complex form of Fourier series • Introduction to Fourier transforms and the convolution theorem Blue book New chapter 12 Dr Mervyn Roy (S6)

  2. Last time Any function in the range can be represented by the Fourier sine series Periodic extension of Fourier sine series - represent waves with odd symmetry Use Fourier series formula to find unknown coefficients when solving the wave equation

  3. “ One point … cannot be emphasized too strongly. To use mathematics effectively in problems, you need not just knowledge, but skill. Skill can be obtained only through practice. “ - Mary L. Boas

  4. Use Fourier methods to find solutions to other PDEs e.g. Laplace equation (see workshop 1 exercise 3), Imagine the boundary conditions are and then and can find coefficients from boundary condition for

  5. Obtaining series expressions for constants - see workshop 1 exercise 1 True for all Try some values and see what happens (e.g. ) Expansion for  first discovered around 1400 by Indian mathematician, Madhava (found  to 11 d.p.).

  6. Fourier cosine series (half range) Within both sines and cosines functions form a mathematically complete set Within this range we can equally well use either sinesor cosines to represent any function

  7. Fourier cosine series A function in the range can be represented by the Fourier cosine series where

  8. Periodic extension of Fourier cosine series We know that cosine waves have even symmetry, and that

  9. triangle wave • wave • exp wave (even)

  10. Comparison of sine and cosine series sine series, slow convergence cosine series, rapid convergence

  11. Comparison of sine and cosine series sawtooth wave sine series, slow convergence triangle wave cosine series, rapid convergence

  12. Full range Fourier series defined within Periodic extension:

  13. oddfull range series same as sine series • evenfull range series same as cosine series • is neither even or odd

  14. Half range, Can expand in terms of sine or cosine functions Full range, In general must use both sines and cosines to represent If is odd all and full range Fourier series same as half range sine series If is even all and full range Fourier series same as half range cosine series

  15. Some physics! square wave n

  16. Some physics! The Swift gamma ray burst mission power spectrum lightcurve

  17. Some physics! Quantum dot 5 nm quantum states, ||2 potential, V

  18. Complex form of Fourier series Full range series: Could equally well write - sometimes the complex form is much more convenient

  19. Complex form of Fourier series

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