3.1. FOURIER TRIGONOMETRIC SERIES
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Using the trig sum formulas, this can be written as ¶ µ ¶¸ Z · µ 1 L 2πx 2πx sin (n + m) + sin (n − m) dx. 2 0 L L
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But this equals zero, because both of the terms in the integrand undergo an integral number of complete oscillations over the interval from 0 to L, which means that the total area under the curve is zero. (The one exception is the (n − m) term if n = m, but in that case the sine is identically zero anyway.) Alternatively, you can simply evaluate the integrals; the results involve cosines whose values are the same at the endpoints of integration. (But again, in the case where n = m, the (n − m) term must be treated separately.) The same kind of reasoning shows that the integral, µ ¶ µ ¶ ¶ µ ¶¸ µ Z · Z L 2πx 1 L 2πx 2πmx 2πnx cos (n + m) cos dx = + cos (n − m) dx, cos L L 2 0 L L 0 (4) equals zero except in the special case where n = m. If n = m, the (n − m) term is identically 1, so the integral equals L/2. (Technically n = −m also yields a nonzero integral, but we’re concerned only with positive n and m.) Likewise, the integral, µ ¶ µ ¶ µ ¶ µ ¶¸ Z L Z · 2πnx 2πmx 1 L 2πx 2πx sin sin dx = − cos (n + m) + cos (n − m) dx, L L 2 0 L L 0 (5) equals zero except in the special case where n = m, in which case the integral equals L/2. To sum up, we have demonstrated the following facts: µ ¶ µ ¶ Z L 2πnx 2πmx sin cos dx = 0, L L 0 µ ¶ µ ¶ Z L 2πnx 2πmx L cos cos dx = δnm , L L 2 0 µ ¶ µ ¶ Z L 2πnx 2πmx L sin sin dx = δnm , (6) L L 2 0 where δnm (known as the Kronecker delta) is defined to be 1 if n = m, and zero otherwise. In short, the integral of the product of any two of these trig functions is zero unless the two functions are actually the same function. The fancy way of saying this is that all of these functions are orthogonal. We normally use the word “orthogonal” when we talk about vectors. Two vectors are orthogonal if their inner product (also called the dot product) is zero, where the inner product is defined to be the sum of the products of corresponding components of the vectors. For example, in three dimensions we have (ax , ay , az ) · (bx , by , bz ) = ax bx + ay by + az bz .
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For functions, if we define the inner product of the above sine and cosine functions to be the integral of their product, then we can say that two functions are orthogonal if their inner product is zero, just as with vectors. So our definition of the inner product of two functions, f (x) and g(x), is Z Inner product ≡
f (x)g(x) dx.
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This definition depends on the limits of integration, which can be chosen arbitrarily (we’re taking them to be 0 and L). This definition is more than just cute terminology. The inner