Explanation From the exercise 23, the Fourier Legendre expansion of the function f ( x ) = x 2 is x 2 = 1 3 P 0 ( x ) + 2 3 P 2 ( x ) . Now, for the function f ( x ) = x 3 , the Fourier-Legendre expansion is as below: The degree of function f ( x ) = x 3 is 3, then equation (1) is written as follows: f ( x ) = c 0 P 0 ( x ) + c 1 P 1 ( x ) + c 2 P 2 ( x ) + c 3 P 3 ( x ) ...... ( 3 ) Here the values of P 0 ( x ) , P 1 ( x ) , P 2 ( x ) and P 3 ( x ) is as follows: P 0 ( x ) = 1 P 1 ( x ) = x P 2 ( x ) = 1 2 ( 3 x 2 − 1 ) P 3 ( x ) = 1 2 ( 5 x 3 − 3 x ) Now from equation (3) it can be written as follows: f ( x ) = c 0 P 0 ( x ) + c 1 P 1 ( x ) + c 2 P 2 ( x ) + c 3 P 3 ( x ) x 3 = c 0 ( 1 ) + c 1 ( x ) + c 2 1

Differential Equations with Boundary-Value Problems

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ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 11.5, Problem 24E

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The Fourier-Legendre series of functions f(x)=x2 and f(x)=x3.

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Chapter 11.5, Problem 23E

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- Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p − 1)/2 multiple of n, i.e. n mod p, 2n mod p, ..., p-1 2 -n mod p. Let T be the subset of S consisting of those residues which exceed p/2. Find the set T, and hence compute the Legendre symbol (7|23). 23 32 how come? The first 11 multiples of 7 reduced mod 23 are 7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8. The set T is the subset of these residues exceeding So T = {12, 14, 17, 19, 21}. By Gauss' lemma (Apostol Theorem 9.6), (7|23) = (−1)|T| = (−1)5 = −1.

Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2 multiple of n, i.e. n mod p, 2n mod p, ..., 2 p-1 -n mod p. Let T be the subset of S consisting of those residues which exceed p/2. Find the set T, and hence compute the Legendre symbol (7|23). The first 11 multiples of 7 reduced mod 23 are 7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8. 23 The set T is the subset of these residues exceeding 2° So T = {12, 14, 17, 19, 21}. By Gauss' lemma (Apostol Theorem 9.6), (7|23) = (−1)|T| = (−1)5 = −1. how come?

Shading a Venn diagram with 3 sets: Unions, intersections, and... The Venn diagram shows sets A, B, C, and the universal set U. Shade (CUA)' n B on the Venn diagram. U Explanation Check A- B Q Search 田

Chapter 11 Solutions

Differential Equations with Boundary-Value Problems

Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...

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The Fourier-Legendre series of functions f ( x ) = x 2 and f ( x ) = x 3 .

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The Fourier-Legendre series of functions f(x)=x2 and f(x)=x3.

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