Question

The polynomial 2x^2 + x + 3 in terms of Legendre polynomials is​

Answer 1

SOLUTION

TO DETERMINE

The polynomial 2x² + x + 3 in terms of Legendre polynomials

EVALUATION

We know that in case of Legendre polynomials :

$\displaystyle \sf{P_0(x) = 1}$

$\displaystyle \sf{P_1(x) = x}$

$\displaystyle \sf{P_2(x) = \frac{1}{2} (3 {x}^{2} - 1) }$

From above we get

$\displaystyle \sf{ {x}^{2} = \frac{2}{3} P_2(x) + \frac{1}{3} }$

Now the given polynomial

$\displaystyle \sf{ = 2 {x}^{2} + x + 3 }$

$\displaystyle \sf{ = 2 \bigg( \frac{2}{3} P_2(x) + \frac{1}{3} \bigg) +P_1(x) + 3 }$

$\displaystyle \sf{ = \frac{4}{3} P_2(x) + \frac{2}{3} +P_1(x) + 3 }$

$\displaystyle \sf{ = \frac{4}{3} P_2(x) + P_1(x) + 3 + \frac{2}{3} }$

$\displaystyle \sf{ = \frac{4}{3} P_2(x) + P_1(x) + \frac{9 + 2}{3} }$

$\displaystyle \sf{ = \frac{4}{3} P_2(x) + P_1(x) + \frac{11}{3} }$

$\displaystyle \sf{ = \frac{4}{3} P_2(x) + P_1(x) + \frac{11}{3} P_0(x) }$

Which is the required expression in terms of Legendre polynomials

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Answer 2

Step-by-step explanation:

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