Laplace Transform Of Periodic Functions
Laplace Transform Of Periodic Functions : Definition : A function f(t) is said to be periodic , with a period ' l ' , if it satisfies the functional equation f(t±l) = f(t) So, if t>0 , a periodic function f(t) can be written as f(t) = f(t + nl) , n= 0,1,2.. ......(1) For example sin t = sin(t+2πn) , n=0,1,2.... is a periodic function with period 2π . In case a function f(t) is periodic , the Laplace transform can be expressed as an integral over one cycle of the function instead of an integral over an infinite range. The transform of equation(1) is ∞ f̅(p) = ∫ e⁻ᵖᵗ f(t) dt a l 2l 3l = [ ∫ + ∫ + ∫ .....]e⁻ᵖᵗ f(t) dt 0 l 2l ∞ (n+1)l = Σ ∫ e⁻ᵖᵗ f(t) dt ............(2)