Problem 17 - Advanced Engineering Mathematics

Question

The first three Legendre polynomials are \(P_{0}(x)=1, P_{1}(x)=x\), and \(P_{2}(x)=\frac{1}{2}\left(3 x^{2}-1\right)\). If \(x=\cos \theta\), then \(P_{0}(\cos \theta)=1\) and \(P_{1}(\cos \theta)=\cos \theta\). Show that \(P_{2}(\cos \theta)=\frac{1}{4}(3 \cos 2 \theta+1)\).

Step-by-Step Solution

Answer

\(P_{2}(\cos \theta) = \frac{1}{4}(3\cos 2\theta + 1)\) is verified.

Step 1: Understand the problem

We need to verify the given equation for the Legendre polynomial when \(x = \cos \theta\). The task is to simplify \(P_{2}(\cos \theta)\) and check if it matches \(\frac{1}{4}(3 \cos 2\theta + 1)\).

Step 2: Substitute \(x=\cos \theta\) into \(P_2(x)\)

Substituting \(x=\cos \theta\) in the polynomial \(P_2(x) = \frac{1}{2}(3x^2 - 1)\), we get:\[P_{2}(\cos \theta) = \frac{1}{2}(3(\cos \theta)^2 - 1).\]

Step 3: Use the double angle identity

We know the double angle formula for cosine: \(\cos 2\theta = 2\cos^2\theta - 1\). We solve for \(\cos^2\theta\) obtaining:\[\cos^2\theta = \frac{1 + \cos 2\theta}{2}.\]

Step 4: Substitute \(\cos^2\theta\) in the expression

Replace \(\cos^2\theta\) in the expression \(P_{2}(\cos \theta) = \frac{1}{2}(3\cos^2\theta - 1)\) to get:\[P_{2}(\cos \theta) = \frac{1}{2}\left(3\left(\frac{1 + \cos 2\theta}{2}\right) - 1\right).\]

Step 5: Simplify the expression

Simplify the equation:\[P_{2}(\cos \theta) = \frac{1}{2}\left(\frac{3(1 + \cos 2\theta)}{2} - 1\right) = \frac{1}{2}\left(\frac{3 + 3\cos 2\theta - 2}{2}\right).\]This simplifies to:\[P_{2}(\cos \theta) = \frac{1}{4}(3\cos 2\theta + 1).\]

Step 6: Conclude

We have shown that \(P_{2}(\cos \theta) = \frac{1}{4}(3\cos 2\theta + 1)\) holds true, as required.

Key Concepts

Double Angle Identities

Understanding double angle identities is crucial when working with Legendre polynomials and trigonometric expressions. These identities allow us to express trigonometric functions of doubled angles, like \(2\theta\), in terms of functions of \(\theta\) itself.

For instance, the double angle identity for cosine is:

• \(\cos 2\theta = 2\cos^2\theta - 1\)

This identity is particularly useful for transforming expressions involving \(\cos^2\theta\) into a simplified form that's often easier to work with. In our exercise, this allowed us to take \(P_{2}(\cos \theta)\) and reorganize it using the identity, making the problem significantly more manageable.

Using double angle identities simplifies the calculations by converting squared trigonometric expressions, which facilitates the manipulation of terms back into a more functional form. As a result, understanding these identities is essential for effectively solving problems involving trigonometric substitutions.

Polynomial Expressions

Polynomials are mathematical expressions consisting of variables and coefficients, sealed by operations like addition, subtraction, and multiplication. In particular, Legendre polynomials are a series of orthogonal polynomials that have applications in physics and engineering.

In the exercise, the specific polynomial expression of interest is \(P_{2}(x) = \frac{1}{2}(3x^2 - 1)\). This polynomial is quadratic, as indicated by the term \(3x^2\). When substituting the variable \(x\) for a trigonometric function such as \(\cos \theta\), the polynomial takes on a new form but maintains its symmetry and properties.

Polynomial expressions can be directly manipulated through algebraic transformations, namely, substitution and simplification, to better understand the behavior of the expression or to fit within a desired calculation format. Mastering the manipulation of polynomial expressions is beneficial when handling more intricate mathematical problems in various scientific fields.

Trigonometric Substitution

Trigonometric substitution is a method used to simplify complex algebraic equations or integrals by replacing variable expressions with trigonometric functions. By converting variable terms into known trigonometric identities, many problems become more straightforward to solve.

In our exercise, trigonometric substitution was performed by assuming \(x = \cos \theta\). This change effectively links the polynomial expressions to trigonometric identities, thereby making the solving process simpler.

Here's the key idea in this kind of substitution:

• Find known relationships or identities for the trigonometric function, such as \(\cos 2\theta = 2\cos^2\theta - 1\), to simplify the resulting equations.

This substitution is particularly advantageous when solving equations involving squared trigonometric terms or when the initial expressions derive from algebraic identities related to trigonometry. By using trigonometric identities, we can streamline the solving process and arrive at the correct forms efficiently.