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Chapter 12: Q 8-4P (page 562)

Find the norm of each of the following functions on the given interval and state the normalized functione-X2/2on(-鈭�,鈭�)

Short Answer

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Step by step solution

01

Concept of normalization factor 

02

Calculate the norm of the function

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Most popular questions from this chapter

We obtained (19.10) forJp(x),p鈮�0.It is, however, valid forp鈮�-1, that is forNp(x),0鈮�p<1. The difficulty in the proof occurs just after (19.7); we said that u,v,u',v'are finite at x=0which is not true forNp(x).

However, the negative powers of x cancel ifp<1. Show this for p=12by using two terms of the power series (12.9) or (13.1) for the function N1/2(x)=-J-1/2(x) [see (13.3)].

Determine the raising and lowering operators for the spherical Bessel functions

Rnjn(x)=nx鈭�Djn(x).

Find the norm of each of the following functions on the given interval and state the normalized functionXe-X2/2on(0,鈭�)

Prove the least squares approximation property of Legendre polynomials [see (9.5) and (9.6)] as follows. Let f(x) be the given function to be approximated. Let the functions pl(x)be the normalized Legendre polynomials, that is, pl(x) = 鈭�(2l+1)/2 Pl(x) , so that

鈭�-11[pl(x)"]"2dx=1.

Show thatLegendre series for f(x)as far as the p2(x)term is

f(x)=c0p0(x)+c1p1(x) +c3p3(x) with c1 =鈭�-11f(x)pl(x) dx

Write the quadratic polynomial satisfying the least squares condition as b0p0(x)+b1p1(x)+b0p0(x)by Problem 5.14 any quadratic polynomial can be written in this form). The problem is to find b0, b1, b2so that I=鈭�-11[f2(x)+(b0-c0)2+(b1-c1)2+(b2-c2)2 -c02 -c12 -c22] dx

Now determine the values of the b's to make I as small as possible. (Hint: The smallest value the square of a real number can have is zero.) Generalize the proof to polynomials of degree n.

Expand the following functions in Legendre series.

f(x) = P'n (x).

Hint: For I鈮� n, 鈭�-11 P'n(x)Pl(x) dx=0 (Why?); for l<n, integrate by parts.
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