Intervals Other Than [-π,π]
Intervals Other Than [-π,π] : So far we have considered the interval [-π,π] only . It was just a matter of convenience , otherwise any finite interval could have been used . We now show that by effecting certain transformations , any finite interval can be made to correspond to the interval [-π,π] . The Interval [0,2π] : If f is bounded , integrable and piecewise monotonic in [0,2π] , then the sum of the series ∞ 1/2 a₀ + Σ (aₙ cos nx + bₙ sin nx) n=1 2π where aₙ = 1/π ∫ f cos nx dx , 0 2π bₙ = 1/π ∫ f sin nx dx 0 is 1/2 [f(x-) + f(x+)] at every point x between 0 and 2π , and is 1/2 [f(2π-) + f(0+)] at x = 0 , 2π and is periodic with period 2π . On substituting x = y+π , we find that y