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mx nx cos dx 0 if m 6 n and L if m n L L mx nx sin dx 0 if m 6 n and L if m n L L mx nx cos dx 0. Where m and n can assume any positive integer values. L L (3)
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An explanation for calling these orthogonality conditions is given on Page 342. Their application in determining the Fourier coe cients is illustrated in the following pair of examples and then demonstrated in detail in Problem 13.4.
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EXAMPLE 1. To determine the Fourier coe cient a0 , integrate both sides of the Fourier series (1), i.e., L L L Xn 1 a0 nx nxo f x dx an cos dx dx bn sin L L L L 2 L n 1 L Now a0 dx a0 L; L 2 L sin
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nx 1 dx 0, therefore, a0 L L
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x EXAMPLE 2. To determine a1 , multiply both sides of (1) by cos and then integrate. Using the orthogonality L 1 L x f x cos dx. Now see Problem 13.4. conditions (3)a and (3)c , we obtain a1 L L L
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If L , the series (1) and the coe cients (2) or (3) are particularly simple. case has the period 2.
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DIRICHLET CONDITIONS Suppose that (1) (2) f x is de ned except possibly at a nite number of points in L; L f x is periodic outside L; L with period 2L
FOURIER SERIES
[CHAP. 13
f x and f 0 x are piecewise continuous in L; L .
Then the series (1) with Fourier coe cients converges to a b f x if x is a point of continuity f x 0 f x 0 if x is a point of discontinuity 2
!0
Here f x 0 and f x 0 are the right- and left-hand limits of f x at x and represent lim f x  and !0 lim f x  , respectively. For a proof see Problems 13.18 through 13.23.
The conditions (1), (2), and (3) imposed on f x are su cient but not necessary, and are generally satis ed in practice. There are at present no known necessary and su cient conditions for convergence of Fourier series. It is of interest that continuity of f x does not alone ensure convergence of a Fourier series.
ODD AND EVEN FUNCTIONS A function f x is called odd if f x f x . Thus, x3 ; x5 3x3 2x; sin x; tan 3x are odd functions. A function f x is called even if f x f x . Thus, x4 ; 2x6 4x2 5; cos x; ex e x are even functions. The functions portrayed graphically in Figures 13-1(a) and 13-1 b are odd and even respectively, but that of Fig. 13-1(c) is neither odd nor even. In the Fourier series corresponding to an odd function, only sine terms can be present. In the Fourier series corresponding to an even function, only cosine terms (and possibly a constant which we shall consider a cosine term) can be present.
HALF RANGE FOURIER SINE OR COSINE SERIES A half range Fourier sine or cosine series is a series in which only sine terms or only cosine terms are present, respectively. When a half range series corresponding to a given function is desired, the function is generally de ned in the interval 0; L [which is half of the interval L; L , thus accounting for the name half range] and then the function is speci ed as odd or even, so that it is clearly de ned in the other half of the interval, namely, L; 0 . In such case, we have 8 L > > a 0; b 2 f x sin nx dx for half range sine series > n n < L 0 L 4 L > 2 nx > > dx for half range cosine series f x cos : bn 0; an L 0 L
PARSEVAL S IDENTITY If an and bn are the Fourier coe cients corresponding to f x and if f x satis es the Dirichlet conditions. Then (See Problem 13.13.) 1 L L