Advanced Calculus fi..

472 Advanced Calculus, Fifth EditionFigure 7.4 Periodic extension of a function defined between -n and n; 7 * -at the discontinuity XI. One cannot expect the series to converge uniformly nearthe discontinuity (cf. Section 6.14), but it will converge uniformly in each closedinterval containing no discontinuity.Although we have up to this point considered only periodic functions f (x)(withperiod 2n), it must be remarked that the basic coefficient formulas (7.9) use only thevalues off (x) between -IT and JT . Thus if f (x) is given only in this interval and is,for example, continuous, then the corresponding Fourier series can be formed, andwe continue to call the series the Fourier series of f (x). If the series converges tof (x) between -n and n, then it will converge outside this interval to a function F(x),which is the "periodic extension of f (x)"; this is illustrated in Fig. 7.4. It shouldbe noted that unless f (n) = f (-n), the process of extension will introduce jumpdiscontinuities at x = n and x = -n. At these points the series will converge to thenumber midway between the two "values" of F(x).We term a function f (x), defined for a 5 x 5 b, piecewise continuous in thisinterval if the interval can be subdivided into a finite number of subintervals, insideeach of which f (x) is continuous and has finite limits at the left and right endsof the interval. Accordingly, inside the ith subinterval the function f (x) coincideswith a function fi(x) that is continuous in the closed subinterval; if, in addition, thefunctions fi(x) have continuous first derivatives, we term f (x) piecewise smooth;if, in addition, the functions f, (x) have continuous second derivatives, we term f (x)piecewise very smooth.FUNDAMENTAL THEOREM Let f (x) be piecewise very smooth in the interval-n ( x 5 n . Then the Fourier series of f (x):RW+ Z(U,,2 n=lon = 1 J f(x) cos nx dx,cos nx + bn sinnx),b,, = Jr f(x) sinnx dx.-R -Rconverges to f (x) wherever f (x) is continuous inside the interval. The series convergesto'[ 2 x-bx,- lim f(x)+ lim f(x)]x+x1+at each point of discontinuity xl inside the interval, and to1-[2 x-+~-lim f(x)+X-b-R+lim f(x)]