10.4we - 10.4: EVEN AND ODD FUNCTIONS KIAM HEONG KWA 1. The...

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10.4: EVEN AND ODD FUNCTIONS KIAM HEONG KWA 1. The Fourier Cosine and Sine Series It is often desired to expand in a Fourier series of period 2 L a func- tion f ( x ) originally defined only on either one of the intervals (0 ,L ), (0 ,L ], [0 ,L ), and [0 ,L ]. One way this can be done is the following. We first extend f ( x ) to a new function onto either one of the intervals ( - L,L ), ( - L,L ], [ - L,L ), and [ - L,L ]. Then the new function is ex- tended periodically onto the real line R . Of particular simplicity and applicability to us later are the following two extensions: The even extension of f ( x ) . Let (1.1) g ( x ) = ( f ( x ) if 0 x L, f ( - x ) if - L < x < 0 . Then we require that g ( x + 2 L ) = g ( x ) for all x R . The function g ( x ) is called the even extension of f ( x ). The odd extension of f ( x ) . Let (1.2) h ( x ) = f ( x ) if 0 < x < L, 0 if x = 0 ,L, - f ( - x ) if - L < x < 0 . Then we require that
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10.4we - 10.4: EVEN AND ODD FUNCTIONS KIAM HEONG KWA 1. The...

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