8. We define the polynomials To(x), T1(x), Ta(x), . . . by the following properties: (a)...
Chapter 2. Legendre Polynomials Examples Show that each function set is orthogonal in the given interval with respect to the specified weight function a. {sin mx}, (-7,7], w(x) = 1 b. {1, 2, 3 (3x2 - 1)}, [-1, 1], w(x) = 1 c. {1, 1 – 2, 3 (x2 - 4x + 2)}; (0,00), w(x) = e-6 Theorem: If the set of functions {P(x)} is orthogonal, then any piece-wise contin- uous function in [a, b] can be represented by the...
We are interested in the first few Taylor Polynomials for the function f(x) = 2e+ + 5e-1 centered at a = 0. To assist in the calculation of the Taylor linear function, T1(x), and the Taylor quadratic function, T2(x), we need the following values: f(0) = 0 f'(0) = f''(0) = 0 Using this information, and modeling after the example in the text, what is the Taylor polynomial of degree one: T1(x) = Preview What is the Taylor polynomial of...
Please explain the solution and write clearly for nu, ber 25.
Thanks.
25. Approximate the following functions f(x) as a linear combination of the first four Legendre polynomials over the interval [-1,1]: Lo(x) = 1, Li(x) = x, L2(x) = x2-1. L3(x) = x3-3x/5. (a) f(x) = X4 (b) f(x) = k (c) f(x) =-1: x < 0, = 1: x 0 Example 8. Approximating e by Legendre Polynomials Let us use the first four Legendre polynomials Lo(x) 1, Li(x)...
4. For tER, we define ft : RR by file) = { if ( 2 ) (a) Show that ft can be written as a power series about r = 0, which converges in some interval (-r,r). (b) For 2 <r, we define P.(t) (n = 0,1,2,...) by fo(t) = ŽP.com Explain why R, P.(t) =et R P.0 and use this to conclude that Pu() = (2) M(0) +. In particular, each P is a polynomial of degree n. These...
From Arfken, obtain recurrence relations for Laguerre
polynomials as mentioned in the text.
By differentiating the generating function in Eq. (13.56) with respect to x and z, we obtain recurrence relations for the LaguerTe polynomials as follows. Using the product rule for differentiation we verify the identities ag ag (13.61) g(x, z)= 2 n=0
By differentiating the generating function in Eq. (13.56) with respect to x and z, we obtain recurrence relations for the LaguerTe polynomials as follows. Using the...
An important fact we have proved is that the family (enr)nez is orthonormal in L (T,C) and complete, in the sense that the Fourier series of f converges to f in the L2-norm. In this exercise, we consider another family possessing these same properties. On [-1, 1], define dn Ln)-1) 0, 1,2, Then Lv is a polynomial of degree n which is called the n-th Legendre polynomial. (a) Show that if f is indefinitely differentiable on [-1,1], thern In particular,...
Q6 (4+3+3+ 6=16 marks) Let Xo, X1, X2 be three distinct real numbers. For polynomials p(x) and q(x), define < p(x),q(x) >= p(xo)q(x0) + p(x1)q(x1) + p(x2)q(22). Let p(n) denote the vector space of all polynomials with degree more no than n. (i) Show that < .. > is an inner product in P(2). (ii) Is < ... > an inner product in P(3)? Explain why. (iii) Is <,:> an inner product in P(1)? Explain why. (iv) Consider Xo =...
Answer True or False and explain
1 The infinite family {Pn(x)}^=o of Legendre polynomials Pn(x) forms a complete orthogonal family on the interval [-1, 1]. If we delete the first element Po(x) = 1 from the set, the remaining family {Pn(x)}=1 also forms a complete orthogonal set. 2 Let {Xn}n=1 be a complete orthogonal family of functions for the vector space L[0, 1]. Then enlarging the set by adding to this set the vector 2X5 + 3X18, we end up...
ou Problem 10.4.3. Show that the first four Hermite polynomials are im Ho = Hi = 2y (10.4.35) (10.4.36) H2 = -2(1 – 2y?) H3 = -12(y - 3) (10.4.37) (10.4.38) where the overall normalization (choice of ao or ai) is as per some convention we need not get into. To compare your answers to the above, choose the starting coefficients to agree with the above. Show that oo e-yHn(y)Hm(y)dy = dnm (VT2"n!) (10.4.39) I-oo for the cases m,n <...
1. (Taylor Polynomial for cos(ax)) For f(x)cos(ar) do the following. (a) Find the Taylor polynomials T(x) about 0 for f(x) for n 1,2,3,4,5 (b) Based on the pattern in part (a), if n is an even number what is the relation between Tn (x) and TR+1()? (c) You might want to approximate cos(az) for all in 0 xS /2 by a Taylor polynomial about 0. Use the Taylor polynomial of order 3 to approximate f(0.25) when a -2, i.e. f(x)...