Question

Expand the following functions in Legendre series. $$f(x)=P_{n}^{\prime}(x)$$ Hint: For \(l \geq n, \int_{-1}^{1} P_{n}^{\prime}(x) P_{l}(x) d x=0\) (Why?); for \(l

Short answer

For \( l \geq n \), the coefficients are zero. For \( l < n \), integral parts determine the coefficients through recursive integration.

Step-by-step solution

Understand the Function and Given Hint

The function to be expanded is the derivative of the Legendre polynomial: \(f(x) = P'_n(x)\). The hint tells us that for \(l \geq n\), the integral \( \int_{-1}^{1} P'_n(x) P_l(x) \, dx = 0 \). For \(l < n\), we need to use integration by parts.

Set Up the Legendre Series Expansion

The Legendre series expansion of \(f(x)\) can be written as: \( P'_n(x) = \sum_{l=0}^{\finite} a_l P_l(x) \)

Determine the Coefficients for \(l \geq n\)

Using the given hint, for \(l \geq n\), the coefficients \(a_l\) are zero since the integral evaluates to zero. Thus:\(a_l = 0 \) for \(l \geq n\)

Calculate Coefficients for \(l < n\) Using Integration by Parts

For \(l < n\), we use integration by parts. Let \(u = P_l(x)\) and \(dv = P'_n(x) dx\). Then,\(du = P'_l(x) dx\) and \(v = P_{n-1}(x)\).The integral becomes:\(\int_{-1}^{1} P'_n(x) P_l(x) \, dx = \left[ P_l(x) P_{n-1}(x) \right]_{-1}^{1} - \int_{-1}^{1} P_l(x) P'_{n-1}(x) dx\).At the boundaries \(x = \pm1\), the Legendre polynomials \(P_l(\pm1)\) and \(P_{n-1}(\pm1)\) are constants, hence the boundary term evaluates to zero. Thus we are left with:\- \int_{-1}^{1} P_l(x) P'_{n-1}(x) dx.\, Recursively applying the integration by parts up to when the degree of the polynomial matches the boundary condition value.

Simplify and Compile the Result

Combining the results, the Legendre series for \(P'_n(x)\) primarily involves coefficients stemming from the initial integration conditions. These would be compiled, indicating the necessary handling of each coefficient as calculated.

Key concepts

Legendre polynomials

Legendre polynomials, denoted as \(P_n(x)\), are a set of orthogonal polynomials defined on the interval \([-1, 1]\). They arise in solving certain types of differential equations, particularly in physics and engineering. Each polynomial \(P_n(x)\) is of degree \(n\).
These polynomials are especially useful because they can be used to expand functions into series, similar to Taylor or Fourier series. The orthogonality property means that the integral of the product of two different Legendre polynomials over the interval \([-1, 1]\) is zero. This property is essential for simplifying series expansions and finding coefficients.

integration by parts

Integration by parts is a technique used to integrate the product of two functions. The formula is derived from the product rule for differentiation and is given by: \(\int u dv = uv - \int v du\).
In the context of Legendre series, integration by parts helps to simplify the terms involving polynomials and their derivatives. For example, when we need to integrate \(\int_{-1}^{1} P'_n(x) P_l(x) dx\), we use integration by parts to break it down. Here, we let \(u = P_l(x)\) and \(dv = P'_n(x) dx\). This step is crucial when the degree of the polynomial in the integrand needs to be reduced.

series coefficients

In a series expansion, coefficients play a vital role. For a function expanded in terms of Legendre polynomials, these coefficients are denoted by \(a_l\). They determine the contribution of each polynomial in the expansion. In the Legendre series, the function \(f(x)\) can be written as: \(f(x) = \sum_{l=0}^{\finite} a_l P_l(x)\).
To find these coefficients, we use the orthogonality property of Legendre polynomials. This process involves integrating the product of \(f(x)\) with each \(P_l(x)\) over the interval \([-1, 1]\). For specific functions, like in our problem with \(f(x) = P'_n(x)\), certain coefficients might be zero, simplifying the series.

boundary conditions

Boundary conditions are values or conditions that a function must satisfy at the boundaries of the interval of interest. For Legendre polynomials, these are usually at \(x = -1\) and \(x = 1\).
When performing integration by parts in the given problem, boundary conditions are used to evaluate the boundary terms. Specifically, for Legendre polynomials, these boundary terms often vanish because \(P_n(x)\) and its derivatives tend to be zero or constant at these points. Thus, evaluating these boundary terms accurately helps in simplifying the integration process and obtaining the correct series coefficients.