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section103 - Boyce/DiPrima 9th ed Ch 10.3 The Fourier...

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1 Boyce/DiPrima 9 th ed, Ch 10.3: The Fourier Convergence Theorem In Section 10.2 we show that if a Fourier series expansion converges and thereby defines a function f , then f is periodic with period 2 L , with the coefficients a m and b m given by In this section we begin with a periodic function f of period 2 L that is integrable on [- L , L ]. We compute a m and b m using the formulas above and construct the associated Fourier series. The question is whether this series converges for each x , and if so, whether its sum is f ( x ) . = + + = 1 0 sin cos 2 ) ( m m m L x m b L x m a a x f π π dx L x m x f L b dx L x m x f L a L L m L L m = = π π sin ) ( 1 , cos ) ( 1 1 Fourier Series Representation of Functions To guarantee convergence of a Fourier series to the function from which its coefficients were computed, it is essential to place additional conditions on the function. From a practical point of view, such conditions should be broad enough to cover all situations of interest, yet simple enough to be easily checked for particular functions. To this end, we recall from Chapter 6.1 the definition of a piecewise continuous function on the next slide. 2 Piecewise Continuous Functions A function f is piecewise continuous on an interval [ a , b ] if this interval can be partitioned by a finite number of points a = x 0 < x 1 < … < x n = b such that (1) f is continuous on each ( x k , x k +1 ) The notation f ( c +) denotes the limit of f (x) as x c from the right, and f ( c -) denotes the limit of f ( x ) as x c from the left.
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