Advanced Calculus fi..

Chapter 7 Fourier Series and Orthogonal Functions 51 9The form (7.82) has various advantages. The coefficients c, can be defineddirectly in terms of f (x):for the integral onInthe right is interpreted asf (x)(cos nx - i sin nx) dx2lr -=See Section 8.2 for the theory of such integrals of complex funcii'dns. ~ h en' i = 0,this gives iao; when n > 0, the integral equals $(a, - ib,); and when n < 0, it equals$(a_,, + ib-,). One has thus the concise statement,whenever the series converges to f (x).For formal work with Fourier series, and even for computation of coefficients,the series (7.84) provides a considerable simplification.By a suitable limiting process the Eqs. (7.84) lead to the relations11 O0f (x) = -/0° g(t)eix' dt. g(t) = -/ f (x)eFiXt dx . (7.85)2/27; -00,fz -ccThus under appropriate hypotheses a function f (x) defined for -cc < x < cc canbe represented as a "continuous sum" of sines and cosines (el"' = cos xt + i sin xt).The integral representing f (x) is termed the Fourier integral of f (x). The Fouriercoefficients off (x) in this integral representation are the numbers g(t), which forma new function. Equations (7.85) show that the relation between f and g is nearlysymmetrical.Equations (7.85) can also be written in real form as follows:f (I) = jo a(r) cosxt dr + lo B(t) sinxr dr.dod ~ , ~ ( 1 = ) 1 /Scca(r) = - ~(X)COSXI-cc -,f (x)sinxr dx.(7.86)This is directly analogous to the real form for Fourier series.The validity of the formulas (7.85) or the equivalent formulas (7.86) can beestablished if f (x) is piecewise smooth in each finite interval and the integral