Engineering Mathematics Class # 14 Fourier Series, Integrals, and Transforms (Part 2)

1. Engineering Mathematics Class #14Fourier Series, Integrals, and Transforms (Part 2) Sheng-Fang Huang

2. 11.3 Even and Odd Functions. Half-Range Expansions • The g is even if g(–x) = g(x), so that its graph is symmetric with respect to the vertical axis. • A function h is odd if h(–x) = –h(x). • The function is even, and its Fourier series has only cosine terms. The function is odd, and its Fourier series has only sine terms.

3. Fig. 263. Odd function Fig. 262. Even function

4. THEOREM 1 The Fourier series of an even function of period 2L is a “Fourier cosine series” (1) with coefficients (note: integration from 0 to L only!) (2) Fourier Cosine Series

5. THEOREM 1 The Fourier series of an odd function of period 2L is a “Fourier sine series” (3) with coefficients (4) Fourier Sine Series

6. THEOREM 2 Sum and Scalar Multiple The Fourier coefficients of a sum ƒ1 + ƒ2 are the sums of the corresponding Fourier coefficients of ƒ1 and ƒ2. The Fourier coefficients of cƒ are c times the corresponding Fourier coefficients of ƒ.

7. Example 1: Rectangular Pulse • The function ƒ*(x) in Fig. 264 is the sum of the function ƒ(x) in Example 1 of Sec 11.1 and the constant k. Hence, from that example and Theorem 2 we conclude that

8. Example 2: Half-Wave Rectifier • The function u(t) in Example 3 of Sec. 11.2 has a Fourier cosine series plus a single term v(t) = (E/2) sin ωt. We conclude from this and Theorem 2 that u(t) – v(t) must be an even function. u(t) – v(t) with E = 1, ω = 1

9. Example 3: Sawtooth Wave • Find the Fourier series of the function ƒ(x) = x + π if –π <x < π and ƒ(x + 2π) = ƒ(x).

10. Solution.

11. Half-Range Expansions • Half-range expansions are Fourier series ( Fig. 267). • To represent ƒ(x) in Fig. 267a by a Fourier series, we could extend ƒ(x) as a function of period L and develop it into a Fourier series which in general contain both cosine and sine terms.

12. Half-Range Expansions • For our given ƒ we can calculate Fourier coefficients from (2) or from (4) in Theorem 1. • This is the even periodic extension ƒ1 of ƒ (Fig. 267b). If choosing (4) instead, we get (3), the odd periodic extension ƒ2 of ƒ (Fig. 267c). • Half-range expansions: ƒ is given only on half the range, half the interval of periodicity of length 2L. 493

13. Fig. 267. (a) Function ƒ(x) given on an interval 0 ≤x ≤L

14. Fig. 267. (b) Even extension to the full “range” (interval) –L ≤x ≤L (heavy curve) and the periodic extension of period 2L to the x-axis

15. Fig. 267. (c) Odd extension to –L ≤x ≤L (heavy curve) and the periodic extension of period 2L to the x-axis

16. Example 4: “Triangle” and Its Half-Range Expansions • Find the two half-range expansions of the function (Fig. 268)

17. Solution. (a) Even periodic extension.

18. Solution. (b) Odd periodic extension.

19. Fig. 269. Periodic extensions of ƒ(x) in Example 4

20. 11.4 Complex Fourier Series. • Given the Fourier series can be written in complex form, which sometimes simplifies calculations. This complex form can be obtained by the basic Euler formula

21. Complex Fourier Coefficients • The cnare called the complex Fourier coefficients of ƒ(x). (6) • For a function of period 2L our reasoning gives the complex Fourier series (7)

22. Example 1: Complex Fourier Series • Find the complex Fourier series of ƒ(x) = exif –π < x < π and ƒ(x + 2π) = ƒ(x) and obtain from it the usual Fourier series. • Solution.

23. Example 1: Complex Fourier Series • Solution.

24. Fig. 270. Partial sum of (9), terms from n = 0 to 50