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)]

                                                                            0

   = 0     for m≠n 

  The infinite set of eigen functions of the Sturm - Liouville Problem (1) and (2) form an orthogonal system with respect to weight function γ , as is evident from the following theorem .

Theorem :-

     If φₘ and φₙ are eigen functions of the Sturm - Liouville Problem(1) and (2) corresponding to the distinct eigen values λₘ and λₙ respectively , then they are orthogonal with respect to the weight function γ .

Proof :

                    Since φₘ is an eigen function   corresponding to λₘ . It satisfies       equation (1) with λ = λₘ . Hence we have,

 d/dx [p(x)φ'ₘ(x)] + [q(x)+λₘ γ(x)] φₘ(x) = 0

Similarly for φₙ we get 

 d/dx [p(x)φₙ'(x)] + [q(x) + λₙ γ(x)]φₙ(x) =0

Multiply equation(15) by φₙ(x) and equation(16) by φₘ(x) and subtracting we obtain 

  (λₘ-λₙ) γ(x) φₘ(x)φₙ(x) = 

                                    d/dx [p(x)φₙ'(x)]φₘ(x) 

                                   -  d/dx [p(x)φₘ'(x)]φₙ(x),

that is 

(λₘ- λₙ) γ(x) φₘ(x) φₙ(x) = 

       d/dx [p(x){φ'ₙ(x) φₘ(x) - φₘ'(x) φₙ(x)}];

 which on integration yields 

                    b

     ( λₘ- λₙ) ∫ γ(x) φₘ(x) φₙ(x) dx

                    a

        = p(b){φ'ₙ(b) φₘ(b) - φ'ₘ(b) φₙ(b)} 

         - p(a){φₙ'(a) φₘ(a) - φ'ₘ(a) φₙ(a)}

    The orthogonality of φₙ and φₘ is established if we show that the right side of equation(17 ) vanishes . 

  From the first equation of (2) we get 

  a₁ φₘ(a) + a₂ φₘ'(a) =0 and a₁φₙ(a) + a₂φₙ'(a)

                                                                     = 0

Elimination of a₂ from these two equations leads to 

  a₁[φₙ(a) φₘ'(a) - φₘ(a) φₙ'(a)] = 0

and if a₁ ≠ 0 then 

             φₙ(a) φ'ₘ(a) - φₘ(a) φ'ₙ(a)=0 ..........(18)

Similarly if b₁≠0 then from 2nd equation of (2) we get 

    φₙ(b) φ'ₘ(b) - φₘ(b) φₙ'(b) = 0 ...............(19)

By (18) and (19) the right side of (17) vanishes, establishing that φₘ and φₙ are orthogonal . 

   If either a₁= 0 or b₁= 0 or a₁= b₁= 0 it can be checked that the right side of(17) vanishes .

Corollary (Eigen function Expansion) :-

    Let g(x) be a piece wise continuous function defined on [a,b] satisfying the boundary conditions (2) . Let φ₁,φ₂,......φₙ be the set of eigen functions of the Sturm - Liouville Problems (1) and (2) . Then 

  g(x) = c₁φ₁(x) + c₂φ₂(x) + ....+cₙφₙ(x)+... ...(20)

 Where cₙ's are given by 

      b                             b

 cₙ ∫ γ(x) φₙ²(x) dx = ∫γ(x) g(x) φₙ(x) dx

     a                              a

                                          n= 1,2,3....  ........(21)

Proof : 

       By the above theorem φ₁,φ₂,...φₙ form an orthogonal system with respect to γ on [a,b] . Hence 

    b

    ∫ γ(x) φₙ(x) φₘ(x) dx = 0 for n≠m  .....(22)

   a      

 From (20)    we have 

b

∫ g(x) γ(x) φₙ(x) dx = 

a   

                 b   ∞

                 ∫(  Σ  cₖ φₖ(x) ) γ(x) φₙ(x) dx   

                 a k=1     

         ∞         b

      = Σ   cₖ (∫ γ(x) φₖ(x) φₙ(x) dx )

        k=1     a  

              b

      = cₙ ∫ γ(x) φₙ²(x) dx , by (22)

              a    

 Here we have assumed the uniform convergence of the series in the right side of (20) .