Orthogonality Of Eigen Functions
Orthogonality Of Eigen Functions :- Definition : Two distinct continuous functions f and φ on [a,b] are said to be orthogonal with respect to a continuous weight function γ if b ∫ f(x) φ(x) γ(x) dx = 0 ...(14) a An infinite set of functions defined on [a,b] is to be an orthogonal system with respect to the weight function γ on [a,b] if every pair of distinct functions of the set are orthogonal with respect to γ . Example : The set of functions {φₙ} , where φₙ(x) = sin nx , n=1,2,.... on [0,π] is an orthogonal system with respect to the weight function having the constant value 1 on [0,π] , for π ∫ (sin mx)(sin nx)(1) dx 0 π =[sin(m-n)x / 2(m-n) - sin(m+n)x / 2(m+n)]