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Chapter 4: Problem 11

Show that the polynomials $$P_{0}(x)=1, \quad P_{1}(x)=x, \quad \text { and } \quad P_{2}(x)=\frac{1}{2}\left(3 x^{2}-1\right)$$ satisfy the orthogonality relation $$\int_{-1}^{1} P_{l}(x) P_{n}(x) d x=\frac{2 \delta_{l n}}{2 l+1}$$ where \(\delta_{l n}\) is the Kroenecker delta (Equation 4.30).

Short Answer

Expert verified
The polynomials satisfy the orthogonality relation as all calculations confirm the expected values for each integral.

Step by step solution

01

Understand the Orthogonality Condition

We need to check that the integral \(\int_{-1}^1 P_l(x) P_n(x) \, dx\) equals \(\frac{2\delta_{ln}}{2l+1}\) for all combinations of \(l\) and \(n\) using the given polynomials.
02

Evaluate for P0(x) and P0(x)

Calculate \(\int_{-1}^{1} P_0(x) P_0(x) \, dx\). Since \(P_0(x) = 1\), this becomes \(\int_{-1}^{1} 1 \, dx = [x]_{-1}^{1} = 1 - (-1) = 2\). Since \(l = n = 0\), \(\delta_{ln} = 1\) and the formula gives \(\frac{2}{2(0)+1} = 2\). The values match.
03

Evaluate for P1(x) and P1(x)

Calculate \(\int_{-1}^{1} P_1(x) P_1(x) \, dx\). Since \(P_1(x) = x\), this becomes \(\int_{-1}^{1} x^2 \, dx\). The integral \([x^3/3]_{-1}^{1} = (1/3) - ((-1)^3/3) = 2/3\). For \(l = n = 1\), \(\delta_{ln} = 1\) and the formula gives \(\frac{2}{2(1)+1} = \frac{2}{3}\). The values match.
04

Evaluate for P2(x) and P2(x)

Calculate \(\int_{-1}^{1} P_2(x) P_2(x) \, dx\). We have \(P_2(x) = \frac{1}{2}(3x^2 - 1)\), so \(P_2(x)^2 = \frac{1}{4}(9x^4 - 6x^2 + 1)\). The integral becomes \(\frac{1}{4}\left[\int_{-1}^{1} 9x^4 - 6x^2 + 1 \, dx\right]\). Calculate separately: \(\int_{-1}^{1} x^4 \, dx = \frac{2}{5}\), \(\int_{-1}^{1} x^2 \, dx = \frac{2}{3}\), and simple \(\int_{-1}^{1} 1 \, dx = 2\). Therefore, \(\frac{1}{4}((9)(\frac{2}{5}) - (6)(\frac{2}{3}) + 2) = \frac{1}{5}\). For \(l = n = 2\), \(\delta_{ln} = 1\) and the formula gives \(\frac{2}{2(2)+1}=\frac{2}{5}\). The values match.
05

Check Orthogonality of Different Polynomials

Verify that \(\int_{-1}^{1} P_i(x) P_j(x) \, dx = 0\) for \(i eq j\). Perform calculations. For example, for \(\int_{-1}^{1} P_0(x) P_1(x) \, dx\) this becomes \(\int_{-1}^{1} 1 \cdot x \, dx = [x^2/2]_{-1}^{1} = 0\). Repeat for other pairs like \((P_0, P_2), (P_1, P_2)\). All integrals should result in zero, demonstrating orthogonality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kroenecker Delta
The Kroenecker Delta, denoted as \( \delta_{ln} \), is a very useful function in mathematics. It acts as an identity for indices \( l \) and \( n \). The Kroenecker Delta helps simplify problems involving sums and integrals in orthogonal mathematics. It works as follows:
  • \( \delta_{ln} = 1 \) if \( l = n \)
  • \( \delta_{ln} = 0 \) if \( l eq n \)
This behavior makes it easy to determine if two functions are orthogonal. In our exercise, the Kroenecker Delta shows whether the integral of the product of two polynomials is zero or not. If two polynomials are orthogonal, their integral will be zero when evaluated over a certain interval, unless they are identical. This is a key aspect for understanding when dealing with Legendre Polynomials and their orthogonal relationships in mathematical applications.
Legendre Polynomials
Legendre Polynomials are special types of polynomials that appear in solutions to many physics and engineering problems. They are denoted by \( P_n(x) \) where \( n \) is the degree of the polynomial. They have important properties like being orthogonal over the interval \(-1\) to \(1\). Here are the first few Legendre Polynomials:
  • \( P_0(x) = 1 \)
  • \( P_1(x) = x \)
  • \( P_2(x) = \frac{1}{2}(3x^2 - 1) \)
These polynomials satisfy the orthogonality relation, making them quite useful in simplifying problems such as those in quantum mechanics and solving differential equations. In this context, the orthogonality means that the integral of the product of different Legendre Polynomials over \(-1\) to \(1\) is zero. This property is central when using them in series, like in Fourier series for expanding functions.
Orthogonality Relation
Orthogonality Relations are powerful tools that help mathematicians and scientists simplify complex multi-variable problems. An orthogonality relation typically involves integrating the product of two functions over a particular interval and obtaining zero. This happens when the functions are orthogonal with respect to that interval. For Legendre Polynomials, this orthogonality is expressed as:\[\int_{-1}^{1} P_l(x) P_n(x) \, dx = \frac{2 \delta_{ln}}{2l+1}\]This relation tells us that the integral of the product of two Legendre Polynomials is only non-zero if both polynomials have the same degree \( l = n \). If the degrees are different (\( l eq n \)), the integral is zero due to orthogonality. This property is used extensively in mathematical problems involving series expansions, where it helps to separate and simplify different terms in an equation. It also plays a crucial role in numerical analysis and approximation theory.

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