a) Show that the orthogonality condition (44) for Legendre

Chapter 8, Problem 31E

(choose chapter or problem)

(a) Show that the orthogonality condition (44) for Legendre polynomials implies that

\(\int_{-1}^{1} P_{n}(x) q(x) d x=0\)

for any polynomial \(q(x)\) of degree at most \(n-1\).. [Hint: The polynomials \(P_{0}, P_{1}, \ldots, P_{n-1}\) are linearly independent and hence span the space of all polynomials of degree at most \(n-1\). Thusq, \((x)=a_{0} P_{0}(x)+\ldots+a_{n-1} P_{n-1}(x)\) for suitable constants \(a_{k}\).]

(b) Prove that if \(Q_{n}(x)\) is a polynomial of degree \(n\) such that

\(\int_{-1}^{1} Q_{n}(x) P_{k}(x) d x=0\)

for \(k=0,1, \ldots, n-1\), then \(Q_{n}(x)=c P_{n}(x)\)

for some constant \(c\).

[Hint: Select \(c\). so that the coefficient of \(x^{n}\)for \(Q_{n}(x)=c P_{n}(x)\) is zero. Then, since \(P_{0}, \ldots, P_{n-1}\)  is a basis, \(Q_{n}(x)=c P_{n}(x)=a_{0} P_{0}(x)+\ldots+a_{n-1} P_{n-1}(x)\)

Multiply the last equation by \(P_{k}(x)\)

\((0 \leq k \leq n-1)\) and integrate from \(x=-1 \text { to } x=1\) to show that each \(a_{k}\) is zero.]

Equation Transcription:

     

   

     

 

   

     

     

   

   

   

       

   

 

     

         

   

     

Text Transcription:

\in-11Pn(x)q(x)dx=0    

q(x)    

n-1    

P0,P1,...,Pn-1

n-1  

q(x)=a0P0(x)+...+an-1Pn-1(x)    

ak

Qn(x)    

n      

-11Qn(x)Pk(x)dx=0

k=0,1,...,n-1  

Qn(x)=cPn(x)    

c    

c      

xn  

Qn(x)=cPn(x)

P0,...,Pn-1      

Qn(x)=cPn(x)=a0P0(x)+...+an-1Pn-1(x)          

Pk(x)

(0 \leq k \leq n-1)    

x= -1 to x=1    

ak