Fourier Series For Even And Odd Functions
Fourier Series For Even And Odd Functions : Even Function : If f is an even function, i.e , f(-x) =f(x) , ∀ x , then f cos nx is an even and f sin nx is an odd function and therefore π aₙ = 1/π ∫ f cos nx dx -π 0 π = 1/π [∫ f cos nx dx + ∫ f cos nx dx] -π 0 π = 2/π ∫ f cos nx dx .........(1) 0 π bₙ = 1/π ∫ f sin nx dx =0 -π So , the Fourier Series of an even function consists of terms of cosines only and the coefficients aₙ may be computed from equation(1) . Also , since for an even function , f(0+) = f(0-) = f(0) and f(-π+0) = f(π-0) the sum of the series f(0) at 0 or ±(even multiple of π) , and is f(π-) at ±π or ±(odd multiple of