Half Range Series
Half Range Series : With the help of the Main Theorem and those of even and odd functions , we now consider the expansion of a function over the interval [0,π] in terms of (i) sine terms only , (ii) cosine terms only . (i) The Sine Series : If a function f is bounded , integrable and piecewise monotonic in [0,π] , then the sum of the sine series π Σ bₙ sin nx , where bₙ = 2/π ∫f sin nx dx 0 is equal to , 1/2 [f(x-) + f(x+)] at every point x between 0 and π , and is equal to 0 , when x=0 ,π . To obtain a series consisting of only sine terms we define an odd function F in [-π,π] , identical with f in [0,π] . Let F = f in [0,π] , and F(x) = -F(-x) = -f(-x) in [-π,0]. Evidently , F is bounded , integrable and piecewise monotone in [-π,π], (i.e. satisfies the conditions of the