By Hugo. Rossi

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Both the numerator and the denominator are zero there. However we can evaluate c 0 by noting that according to Eq. 37) The series converges to x everywhere. Since cos(nπ x) = cos(−nπ x) it is an even function and its periodic continuation is shown in Fig. 3. Note that the sine series is discontinuous at x = 1, while the cosine series is continuous everywhere. 38) n=1 where cn = 1 x=0 f (x) cos(nπ x)d x 1 2 x=0 cos (nπ x)d x n = 0, 1, 2, 3, . . bn = 1 x=0 f (x) sin(nπ x)d x 1 2 x=0 sin (nπ x)d x n = 1, 2, 3, .

100) requires that D = 0. The second requires that C sin(λ) be zero. Our eigenvalues are again λn = nπ . The boundary condition at τ = 0, that Pτ = 0 requires that A = 0. 3 Orthogonality Applying the final (nonhomogeneous) boundary condition (the initial position). 109) Lessons The solutions are in the form of infinite series. The coefficients of the terms of the series are determined by using the fact that the solutions of at least one of the ordinary differential equations are orthogonal functions.

6th edition. New York: McGraw-Hill, 2001. book Mobk070 March 22, 2007 11:7 book Mobk070 March 22, 2007 11:7 31 CHAPTER 3 Orthogonal Sets of Functions In this chapter we elaborate on the concepts of orthogonality and Fourier series. We begin with the familiar concept of orthogonality of vectors. We then extend the idea to orthogonality of functions and the use of this idea to represent general functions as Fourier series—series of orthogonal functions. Next we show that solutions of a fairly general linear ordinary differential equation—the Sturm–Liouville equation—are orthogonal functions.