Prove the recurrence relation xPn(x) = n + 1 2n + 1 Pn+1(x) + n 2n + 1 Pn−1(x), and evaluate the following integral using the orthogonal property of Legendre polynomials Z 1 −1 xPn(x)Pn−1(x)dx
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A question from (mathematical physics - Couchy integral
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c) The Rodrigues formula of Legendre polynomials can be converted into the Schlafli integral as (-1)" 1 (1 - 22n Pn(x) = dz 2n 2ni (z - x)n+1 C is a closed contour encircles the point z = x C
--Red In( - ) () -2 21 (+1) 2+1(n + m)} ml(m + 2n + 1) (1 - 1) de 22+1(n!) (2n +1) 8. By evaluating ac 2h +G ah where G(h) is the generating function for Legendre polynomials, show that 1 - 2 Σ (2n +1) Po (1 - 2ch + ha) Hence, or otherwise, prove that Pn(x) dx 2h 9. Given that {(2, 2) = ( 12h the hm-dh m>1. prove that 2am+ | 112,0)P.a)dt (m + n)...
3. Let Po(x) respectively. Find the monic Legendre polynomials of degree 2, 3 and 4 using the orthogonality relation f P(x)Pm(x)dx = 0, mn and m,n E N. 1 and P1(x) = x be two Legendre polynomials of degree 0 and 1,
3. Let Po(x) respectively. Find the monic Legendre polynomials of degree 2, 3 and 4 using the orthogonality relation f P(x)Pm(x)dx = 0, mn and m,n E N. 1 and P1(x) = x be two Legendre polynomials of...
Prove the following integral. ., xPn-1(x)P,(x)dx = 2n (2n-1)(2+1) 2 use (n + 1)Pn+1(x) – (2n +1)xPn(x) + nPn-1(x) = 0, L, Pn(x)Pm (x) dx = 0, S, Pn(x)2 dx = 2n+1
3. Use the recurrence relation to obtain ex ,P(x),P,(z),B(x), assuming that P)(z) = i. Pi (x)-z. Then sketch the graphs of P,.(x) for n-0. Î,2.3.4.5 İn the interval-1-z-i in one Figure. You may use any software to produce the graphs. pressions for the Legendre polynomials P2(r)
3. Use the recurrence relation to obtain ex ,P(x),P,(z),B(x), assuming that P)(z) = i. Pi (x)-z. Then sketch the graphs of P,.(x) for n-0. Î,2.3.4.5 İn the interval-1-z-i in one Figure. You may use...
From Arfken, demostrate equation 12.85. Step by step solution
please.
Associated Legendre Polynomials The regular solutions, relabeled pn (x), are (12.73c) These are the associated Legendre functions.16 Since the highest power of x in Pn (x) is xn, we must have m n (or the m-fold differentiation will drive our function to zero) In quantum mechanics the requirement that m n has the physical interpretation that the expectation value of the square of the z component of the angular momentum...
(a) Use the generating function of the Legendre polynomials: G(u, z) = (i) to determine the oddness or evenness of the Legendre polynomial P) de- pending on n; (ii) to obtain the particular values P) and P-1); (iii) to show that and then to deduce (n + 1)P,n(x) = (2n + 1)2Pa(z) _ nP,-1(x), n-1,2,
(a) Use the generating function of the Legendre polynomials: G(u, z) = (i) to determine the oddness or evenness of the Legendre polynomial P) de-...
Problem 4.2.
Please only do 4.2, I only need help with
4.2
Only 4.2 please
. OVERFITTING 4.4 Problems pict the monomials of order i, фі(x-a". As you increase correspond to the intuitive notion of increasing complexity? Prob Problem 4.2 Consider the feature transform z = Lo(2), L1(x), L2(zr and the linear model h(x) w'z. For the hypothesis with w [1,-1, 1 what is h(z) explicitly as a function of z? What is its degree? Problem 4.3 The Legendre Polynomials...
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,...
from arfken
Rework Example 10.3.1 by replacing фп(x) by the conventional Legendre polynomial. Pn (x): [Pn (x)' dx = 2n + 1 Using Eqs. (10.47a), and (10.49a), construct Po, P1(x), and P2(x).