1. Numerical Methods Continuous Fourier Series Part: Continuous Fourier Series http://numericalmethods.eng.usf.edu

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5. http://numericalmethods.eng.usf.edu5 Chapter 11.02: Continuous Fourier Series Lecture # 2 For a function with period “T” continuous Fourier series can be expressed as The “average” function value between the time interval [0,T] is given by (22) (23)

6. http://numericalmethods.eng.usf.edu6 Even and Odd functions are described as (24) (25) Continuous Fourier Series

7. http://numericalmethods.eng.usf.edu7 Derivation of formulas for Integrating both sides with respect to time, one gets The second and third terms on the right hand side of the above equations are both zeros (26)

8. http://numericalmethods.eng.usf.edu8 (27) (28) Derivation of formulas for

9. http://numericalmethods.eng.usf.edu9 Now, if both sides are multiplied by and then integrated (29) Derivation of formulas for

10. http://numericalmethods.eng.usf.edu10 The first and second terms on the RHS of Equation (29) are zero. The third RHS term of Equation (29) is also zero, with the exception when (30) Derivation of formulas for

11. http://numericalmethods.eng.usf.edu11 Similar derivation can be used to obtain Derivation of formulas for

12. THE END http://numericalmethods.eng.usf.edu

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16. Numerical Methods Continuous Fourier Series Part: Example 1 http://numericalmethods.eng.usf.edu

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20. http://numericalmethods.eng.usf.edu20 Chapter 11.02: Example 1 (Contd.) Lecture # 3 Using the continuous Fourier series to approximate the Find the Fourier coefficients and Figure 1: A Periodic Function following periodic function in Fig (1).

21. http://numericalmethods.eng.usf.edu21 Example 1 cont. From Equations (23-25), one obtains :

22. http://numericalmethods.eng.usf.edu22 Example 1 cont.

23. http://numericalmethods.eng.usf.edu23 From this equation, we obtain the Fourier coefficients for = -0.9999986528958207 = -0.4999993232285269 = -0.3333314439509194 = -0.24999804122384547 = -0.19999713794872364 = -0.1666635603759553 = -0.14285324664625462 = -0.12499577981019251 Example 1 cont.

24. http://numericalmethods.eng.usf.edu24 We can now find the values of Example 1 cont. from the following equations,

25. http://numericalmethods.eng.usf.edu25 For the Fourier coefficients can be = -0.6366257003116296 = -5.070352857678721e-6  0 = -0.07074100153210318 = -5.070320092569666e-6  0 = -0.025470225589332522 = -5.070265333302604e-6  0 = -0.012997664818977102 = -5.070188612604695e-6  0 Example 1 cont. computed as

26. http://numericalmethods.eng.usf.edu26 Example 1 conclusion In conclusion, the periodic function f(t) (shown in Figure 1) can be expressed as: where and have already computed For one obtains:

27. http://numericalmethods.eng.usf.edu27

28. THE END http://numericalmethods.eng.usf.edu

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30. For instructional videos on other topics, go to http://numericalmethods.eng.usf.edu/videos/ This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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32. Numerical Methods Continuous Fourier Series Part: Complex Form of Fourier Series http://numericalmethods.eng.usf.edu

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36. http://numericalmethods.eng.usf.edu36 Chapter 11.02 :Complex form of Fourier Series (Contd.) Lecture # 4 Using Euler’s identity (31) (32) and

37. http://numericalmethods.eng.usf.edu37 Thus, the Fourier series can be casted in the following form: (33) (34) Complex form of Fourier Series cont.

38. http://numericalmethods.eng.usf.edu38 Define the following constants: Hence: (35) (36) (37) Complex form of Fourier Series cont.

39. http://numericalmethods.eng.usf.edu39 Using the even, odd (38) properties Equation (37) becomes: Complex form of Fourier Series cont.

40. http://numericalmethods.eng.usf.edu40 Complex form of Fourier Series cont. Substituting Equations (35,36, 38) into Equation (34), one gets:

41. http://numericalmethods.eng.usf.edu41 or (39) Complex form of Fourier Series cont.

42. http://numericalmethods.eng.usf.edu42 Complex form of Fourier Series cont. The coefficient can be computed, by substituting or (40) Equations (24, 25) into Equation (36) to obtain:

43. http://numericalmethods.eng.usf.edu43 Substituting Equations (31, 32) into the above equation, one gets: (41) Thus, Equations (39, 41) are the equivalent complex version of Equations (21, 25). Complex form of Fourier Series cont.

44. THE END http://numericalmethods.eng.usf.edu

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