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(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-...
Legendre polynomial 7. Using the Legendre polynomials given by Px(x) = 2mm. An (x2 - 1)" evaluate (a) [ P3(x)dt (b) | 1-1 P2(2) In(1 - 0)dc Hint: Use integration by parts after computing P2() and P3().
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...
2. Show that the following is true of the Legendre polynomial : Pn(1) = 1 V n=1,2,3,... Hint: Use Rodrigues' Formula and recall that (x² - 1) = (x + 1) (x - 1) so that you can employ the Product Rule from calculus.
Determine whether the following function is a polynomial function. If the function is a polynomial function, state its degree. If it is not tell why not. Write the polynomial in standard for FOR = 8x?-o? Determine whether Fx) is a polynomial or not. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A It is not a polynomial because the variable x is raised to the power, which is not a...
Determine whether the following function is a polynomial function. If the function is a polynomial function, state its degree. If it is not, tell why not. Write the polynomial in stu G(x)=2x-3) (2-3) Determine whether Gix) is a polynomial or not. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. It is a polynomial of degree (Type an integer or a fraction) OB. It is not a polynomial because the variable...
Problem 4.4. Determire the generating function for the following 4.4. Determir e the generating function for the following: Hint. For (b), start with th series ㅗ-1 +z+x2+ , differentiate and multiply by r, then differe:.iate aad mtitiply by a, and finally differentiate and multiply by z one last time.
Newton's method in Python
The polynomial function
Implement the polynomial function. The function computes
where
.
def poly(x, a, b, c, d):
#
# complete
#
pass
The derivative of the polynomial
Implement the derivative function given by
.
def poly_der(x, a, b, c, d):
#
# complete
#
pass
Newton's method
Implement Newton's method by updating the solution guess
iteratively. The function returns a tuple of
.
def newtons_method(a, b, c, d, n, x0):
#
# complete
#
pass...
2. The function Pm (x) is the Legendre function which satisfies the differential equation (1 – x²) drpm +m(m + 1)Pm = 0. Please show that, Pm (x)Pn(x)dx = 0 for men. (25 points)
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).