(a) Use the generating function of the Legendre polynomials: G(u, z) = (i) to determine the oddne...
The expression Φ(x, h)-(1-2xh + h2)-1/2 where |hl < 1 is the generating function for Legendre polynomials. φ(x, h) can be expressed as a sum of Legendre polynomials The function (x, h) = Po(x) + hA(x) + h2Pg(x) + hn (x) The generating function of the Legendre polynomials has some applications in Physics, such as expressing the electric potential at point P due to a charge q. The location of the charge is r with respect to the origin O...
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...
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...
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...
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,...
--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)...
1. Let X have probability generating function Gx (s) and let un generating function U(s) of the sequence uo, u1, ... satisfies P(X > n). Show that the (1- s)U(s) = 1 - Gx(s), whenever the series defining these generating functions converge. 1. Let X have probability generating function Gx (s) and let un generating function U(s) of the sequence uo, u1, ... satisfies P(X > n). Show that the (1- s)U(s) = 1 - Gx(s), whenever the series defining...
Problem 2.16 In this problem we explore some of the more useful theorems (stated without proof) involving Hermite polynomials. (a) The Rodrigues formula says that H (6) = (-1)” (1) . (2.87) Use it to derive H3 and H4. (b) The following recursion relation gives you Hn+1 in terms of the two preceding Hermite polynomials: Hn+1(E) = 2€ H, (E) – 2n Hn-1(5). (2.88) Use it, together with your answer in (a), to obtain Hg and H. (c) If you...
Assume that the generating function of the nonnegative integer random variable ξ is G(S) Find the generating functions for the following sequences (1) an=P{ξ≤n} (2) bn=P{ξ=2n}
Problem Six: Given two polynomials: g(x) = anx" + an-iz"-1 +--+ aix + ao Write a MATLAB function (name it polyadd) to add the two polynomials and returns a polynomial t(x) = g(x) + h(x), whether m = n, m < n or m > n. Polynomials are added by adding the coefficients of the terms with same power. Represent the polynomials as vectors of coefficients. Hence, the input to the function are the vectors: g=[an an-1 ao] and h=[am...