Question

The value of \(\int_{-1}^{1}(2 x+1) P_{3}(x) d x\) where \(P_{3}(x)\) is the third degree Legendre polynomial, is (a) \(\mathbf{1}\) (b) \(-1\) (c) 2 (d) \(0 .\)

Short answer

(d) 0

Step-by-step solution

Identify the Legendre Polynomial

The third-degree Legendre polynomial, denoted as \( P_3(x) \), is \( \frac{5}{2}x^3 - \frac{3}{2}x \). These polynomials are orthogonal over the interval \([-1, 1]\). For any polynomial of degree 3 and an order less than 3 (like \(2x + 1\), which is linear), the integral over this interval will be 0 if multiplied by \(P_3(x)\).

Set Up the Integral

Taking into account the orthogonality property of Legendre polynomials, the integral \( \int_{-1}^{1}(2 x+1) P_{3}(x) d x \) can be directly evaluated because \(2x+1\) is a polynomial of degree 1. Since 1<3, the integral evaluates to zero.

Final Evaluation and Conclusion

Given the orthogonality of Legendre polynomials, the integral simplifies immediately, showing that the multiplication of a third-degree polynomial with a polynomial of lower degree, and integrated over its orthogonal interval, is zero. Thus, the answer is verified as \(0\).

Key concepts

Orthogonality of Polynomials

The concept of orthogonality in polynomials is central when dealing with Legendre polynomials. Orthogonal polynomials are those which, when multiplied together and integrated over a specified interval, such as \([-1, 1]\), equal zero. This is similar to the notion of perpendicular vectors in geometry. Just as perpendicular vectors have a dot product of zero, orthogonal polynomials have an integral product of zero over their interval when their degrees differ.

Legendre polynomials, denoted usually as \(P_n(x)\), are a family of orthogonal polynomials. When they are computed over the range from -1 to 1, they exhibit this orthogonality property. This characteristic simplifies many problems in physics and engineering, as well as in mathematical analysis, because it allows the separation of different types of functions based on their degree.

This principle is why, in the given exercise, the integral of a first-degree polynomial \(2x + 1\) with the third-degree Legendre polynomial \(P_3(x)\) results in zero. The orthogonality property tells us immediately that because the degrees are different, they don’t "overlap" in their interval.

Definite Integrals

Definite integrals calculate the accumulated value of a function over a certain interval. In the exercise, the definite integral \(\int_{-1}^{1}(2x+1) P_{3}(x) dx\) is taken over the interval from \(-1\) to \(1\). This involves integrating the product of a function, \(2x + 1\), and a Legendre polynomial \(P_3(x)\) over this range.

In the context of orthogonal polynomials like Legendre polynomials, the value of such an integral can be quickly evaluated without performing detailed calculations. This is because if one polynomial in the integrand comes from a family of orthogonal polynomials and its degree is greater than the degree of the other polynomial present, the definite integral is known to simplify to zero. This makes problems involving orthogonal polynomials straightforward once their degrees are identified.

The given step-by-step solution made use of this property. By recognizing that \(P_3(x)\) is a third-degree polynomial and that \(2x + 1\) is of lower degree, the integral can be determined as zero simply and directly.

Polynomial Degree

The degree of a polynomial indicates the highest power of its variable, usually represented as \(n\) in \((x^n)\). Higher degree polynomials are generally more complex and have more variables. For instance, \(P_3(x) = \frac{5}{2}x^3 - \frac{3}{2}x\) is a third-degree polynomial, where the term \(x^3\) dictates its classification. Linear polynomials, like \(2x + 1\), are considered first degree.

Understanding polynomial degree is essential in the context of integration problems involving orthogonal polynomials. The degree affects the integral's evaluation, as demonstrated in Legendre polynomial scenarios. If you have a polynomial \(P_n(x)\) of a certain degree \(n\), integrating it with another polynomial of lower degree over the orthogonal range will result in zero. This automatic simplification due to degrees can save significant computation time.

In the original exercise, correctly identifying the degrees of the involved polynomials allowed the students to leverage the orthogonality property. Knowing that \(P_3(x)\) was a third-degree and \(2x + 1\) a first-degree, enabled a quick conclusion about the integration's result, showcasing the usefulness of understanding polynomial degrees.