A First Course on Complex Functions by G. Jameson

By G. Jameson

This publication could be of curiosity to undergraduates taking classes on advanced capabilities.

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Example text

21) as a Fourier cosine integral representation. 23) Finally, if f should be a function defined only on the interval O < oo, we can represent it over this interval by either a Fourier cosine integral ora Fourier sine integral, analogous to the ha/f-range expansions of Fourier series. Consider the following example. 2: Find a Fourier cosine and Fourier sine integral representation of the function (see Fig. 2/Fourier Integral Representations • 45 from which we deduce the sine integral representation 2 ("" = ; Jo f(x) (S -s sin TrS/2) sm.

If f and f' are piecewise continuous functions on every finite interval, and if I~"" if(x)idx < 00 then f(x) = ;1 Joroo I""_""f(t)cos[s(t - x)]dt ds at points x where f is continuous. [f(x+) + f(x-)] of the right-hand and left-hand limits. 10) holds, see E. C. Titchmarsh, Theory of Fourier Integrais, Oxford: Clarendon Press, 1937. t Right-hand and left-hand limits are defined, respectively, by f(x+) = lim f(x + e) e-O+ and f(x-) = lim f(x - e). At points of continuity it follows that f(x-) e-O+ = f(x+) = f(x).

2/Fourier Integral Representations • 45 from which we deduce the sine integral representation 2 ("" = ; Jo f(x) (S -s sin TrS/2) sm. 2 1. By using the result of Eq. 14), show directly that (a) ("" sin s cos s ds = :!!. Jo s 4 (b) ("" sin ax dx = :!!. , Jo x 2 a>0 2. _ ("" cos sx + s sin sx ds f(x) 1r Jo i +1 (b) Verify directly that the above integral representation converges to the value 1/2 at x = O. 3. Find an integral representation for f(x) = {1O, - xZ, lxl < lxi > 1 1 and deduce the value of the integral I= l = o (sin X - XCOS X) cos -x 1 dx x3 2 4.

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