Advanced Calculus fi..

This system can be arranged in a definite order to form a system {#,(x, y)], whichis orthogonal and complete for the rectangle R: -n I x 5 IT, -n 5 y 5 n (seeProblem 1 following Section 7.19). One can also write the Fourier series as a"double Fourier series":{a, sin px sin qy + bpq cos px sin qyThis amounts to a special sort of rearrangement and grouping of the series; if the seriesis absolutely convergent, then the reasoning of Section 6.10 shows that the doubleseries has the same sum as the single series in any order. If f (x, y) has continuousfirst and second derivatives for all (x, y) and is periodic in both variables:then an argument similar to that of Section 7.8 shows that the series is absolutelyand uniformly convergent to f (x, y).*7.17 COMPLEX FORM OF FOURIER SERIESFrom the identityof Section 6.19, one derives the relationselx = COS x '+ i sin x (j = a)1a >A Fourier seriesCOSX =elxelx + - e-lxe-lx2 ' sinx = 2i(7.81)5 + &an cos nx + bn sin nx)2n=lcan hence be written in the formThe summation from -oo to oo is understood to mean an addition of two series:If both series converge, the result is clearly the same as the single series on theleft-hand side of (7.82).