Show that the Legendre polynomials P1 and P2 are orthogonal by explicit integration. Also show that...

Show that the Legendre polynomials P1 and P2 are orthogonal by explicit integration. Also show that when (P2)^ 2 is integrated over the full range of integration, the result is 2 /(2l+1) , where l is the order of the polynomial.

Expert Solution

Colby Messinger answered 2 years ago

Show that the first derivatives of the Legendre polynomials satisfy a self-adjoint differential equation with eigenvalue:...

Show that the first derivatives of the Legendre polynomials satisfy a self-adjoint differential equation with eigenvalue: lamda = n(n+1)-2

in the book, Legendre polynomials are obtained to the degree 5. Normalize polynomials P4 and P5...

in the book, Legendre polynomials are obtained to the degree 5. Normalize polynomials P4 and P5 so that P4(1) = 1 and p5(1) = 1. This is standard normalization and these will be standard Legendre polynomials. Use the recurrence relation for standard Legendre polynomials to find two more (standard) Legendre polynomials. Show your work for credit. Hint: The recurrence relation: (n+1) P_{n+1}(x) = (2n+1) P_{n}(x) - n P_{n-1}(x)}

Two polynomials in the variable x are represented by the coefficient vectors p1 = [6,2,7,-3] and p2 = [10,-5,8].

Two polynomials in the variable x are represented by the coefficient vectors p1 = [6,2,7,-3] and p2 = [10,-5,8]. a. Use MuPAD to find the product of these two polynomials; express the product in its simplest form. b. Use MuPAD to find the numeric value of the product if x = 2.

Which of the following sets of polynomials form a basis for P2 (the space of polynomials...

Which of the following sets of polynomials form a basis for P2 (the space of polynomials of degree at most 2)? Explain. (a) {2 + 2x − x 2 , 1 + x, 3x} (b) {1 − x + 2x 2 , 2 + 4x} 1 (c) {3 + x + x 2 , −1 + x, 5 − x + x 2}

. Consider this hypothesis test: H0: p1 - p2 = < 0 Ha: p1 - p2...

. Consider this hypothesis test: H0: p1 - p2 = < 0 Ha: p1 - p2 > 0 Here p1 is the population proportion of “happy” of Population 1 and p2 is the population proportion of “happy” of Population 2. Use the statistics data from a simple random sample of each of the two populations to complete the following: Population 1 Population 2 Sample Size (n) 1000 1000 Number of “yes” 600 280 a. Compute the test statistic z. b....

1. Consider this hypothesis test: H0: p1 - p2 = < 0 Ha: p1 - p2...

1. Consider this hypothesis test: H0: p1 - p2 = < 0 Ha: p1 - p2 > 0 Here p1 is the population proportion of “happy” of Population 1 and p2 is the population proportion of “happy” of Population 2. Use the statistics data from a simple random sample of each of the two populations to complete the following: Population 1 Population 2 Sample Size (n) 1000 1000 Number of “yes” 600 280 a. Compute the test statistic z. b....

Consider this hypothesis test: H0: p1 - p2 = 0 Ha: p1 - p2 > 0...

Consider this hypothesis test: H0: p1 - p2 = 0 Ha: p1 - p2 > 0 Here p1 is the population proportion of “yes” of Population 1 and p2 is the population proportion of “yes” of Population 2. Use the statistics data from a simple random sample of each of the two populations to complete the following: (8 points) Population 1 Population 2 Sample Size (n) 500 700 Number of “yes” 400 560 Compute the test statistic z. What is...

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Show that the Legendre polynomials P1 and P2 are orthogonal by explicit integration. Also show that...

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