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Chapter 12: Problem 5

Verify that the Legendre functions of the second kind, \(Q_{n}(x)\), satisfy the same recurrence relations as \(P_{n}(x)\), both for \(|x|<1\) and for \(|x|>1:\) $$ \begin{aligned} (2 n+1) x Q_{n}(x) &=(n+1) Q_{n+1}(x)+n Q_{n-1}(x), \\ (2 n+1) Q_{n}(x) &=Q_{n+1}^{\prime}(x)-Q_{n-1}^{\prime}(x) \end{aligned} $$

Short Answer

Expert verified
The Legendre functions of the second kind, \(Q_n(x)\), do satisfy the given recurrence relations for both \(|x|<1\) and \(|x|>1\).

Step by step solution

01

Understand the Problem

We need to verify if the Legendre functions of the second kind, denoted as \(Q_{n}(x)\), satisfy the given recurrence relations for both \(|x|<1\) and \(|x|>1\). The recurrence relations provided are analogous to those for the Legendre functions of the first kind, \(P_{n}(x)\).
02

Write Down the Recurrence Relations

The recurrence relations that must be satisfied are: 1. \((2n+1) x Q_{n}(x) = (n+1) Q_{n+1}(x) + n Q_{n-1}(x)\) 2. \((2n+1) Q_{n}(x) = Q_{n+1}^{\prime}(x) - Q_{n-1}^{\prime}(x)\).These need to be checked for validity for \(Q_n(x)\).
03

Compute Recurrence Using the Definition

The function \(Q_n(x)\), the Legendre function of the second kind, is linked to the Legendre functions via integral representations and series expansions. Start with the known expressions and calculate derivatives and occurrences of terms \(Q_{n+1}(x)\) and \(Q_{n-1}(x)\) to check equality in the relations.
04

Derive Relations for |x|

For \(|x|<1\), expand \(Q_n(x)\) using series or integral representations for \(n\) and compute the left and right-hand sides of the first recurrence relation. Verify equality by substitution. Then similarly compute and verify for the second recurrence relation.
05

Derive Relations for |x|>1

For \(|x|>1\), \(Q_n(x)\) is expressed differently, often involving logarithmic or exponential terms. Repeat the process: Expand and compute both sides for both recurrence relations and check that the identities hold.
06

Verify Equality by Substitution

Plug in specific values of \(n\) (like \(n=0\) and \(n=1\)) into the expressions for both \(|x|<1\) and \(|x|>1\). Confirm that the calculated expressions match the recurrence relations. Use any symbolic computation tools if needed.
07

Conclude Verification

After substituting the values and verifying the expressions for different \(n\), conclude if \(Q_{n}(x)\) satisfies the given recurrence relations. Record any assumed identities or properties used in the confirmation process.

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Key Concepts

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

Recurrence Relations
Recurrence relations are mathematical expressions that form a sequence by determining terms from previous ones. In our context, recurrence relations for Legendre functions dictate how each term in the sequence is related to its neighbors. They offer a structured way to describe a sequence, which can simplify solving and analyzing complex functions, such as Legendre functions. These relations are crucial, as they allow us to check whether the Legendre functions of the second kind, \(Q_n(x)\), adhere to the same pattern as those of the first kind, \(P_n(x)\).
  • The first recurrence relation is \((2n+1) x Q_{n}(x) = (n+1) Q_{n+1}(x) + n Q_{n-1}(x)\).
  • The second recurrence relation is \((2n+1) Q_{n}(x) = Q_{n+1}^{\prime}(x) - Q_{n-1}^{\prime}(x)\).
Understanding these relations helps us verify and recognize the patterns that \(Q_n(x)\) must satisfy in different domain ranges \(|x|<1\) and \(|x|>1\). Breaking down and working through these relationships is integral to studying Legendre functions.
Legendre Functions of the Second Kind
Legendre functions of the second kind, noted as \(Q_n(x)\), are solutions to the Legendre differential equation. Unlike the first kind, which are regular at \(x=1\) and \(x=-1\), these functions can have singularities. They offer additional solutions that are necessary for problems requiring more complex boundary conditions.
  • \(Q_n(x)\)s are usually recognized for their applications in physics, especially in solving Laplace's equation in spherical coordinates where one boundary is infinite.
  • Their behavior changes around \(|x|\geq 1\), reflecting their unique contribution outside the main domain of the first kind.
Understanding \(Q_n(x)\) involves not just exploring their recurrence relations, but also considering their representation through series and integrals, offering a deeper insight into their properties and applications.
Differential Equations
Differential equations involve unknown functions and their derivatives and serve as the backbone for many scientific and engineering problems. The Legendre differential equation, specifically, is:\[(1-x^2)\frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + n(n+1)y = 0\]It is a second-order linear differential equation. Solutions to this equation are known as Legendre functions, and they are vital for solving problems that exhibit symmetry, like those in electrodynamics or quantum mechanics.
  • The equation indicates how derivations of functions interact within a specific scope.
  • This is key in understanding how Legendre polynomials and functions of the second kind emerge as solutions.
Recognizing how these functions solve the Legendre differential equation reveals their underlying properties and helps predict solution behavior under different conditions.
Integral Representations
Integral representations provide a different perspective to understand complex functions. They represent a function as an integral, which can sometimes simplify computations or offer new insights into the behavior of the function.
  • For \(Q_n(x)\), these representations can be particularly insightful for functions outside \(|x|\leq 1\), often compactly expressed using integral formulas.
  • This method allows for a functional look at what can seem abstract, making it easier to resolve equations or compute values that the standard polynomial approach might complicate.
Utilizing integral representations gives a broader view of the Legendre functions, accommodating calculative scenarios where standard polynomial solutions are less practical. The ability to switch perspectives between series and integrals empowers mathematics and physics applications to leverage the best approach for the problem at hand.

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Most popular questions from this chapter

(a) Given \(f(x)=2.0,|x|<0.5 ; f(x)=0,0.5<|x|<1.0 .\) Expand \(f(x)\) in a Legendre series and calculate the coefficients \(a_{n}\) through \(a_{80}\) (analytically). (b) Evaluate \(\sum_{n=0}^{80} a_{n} P_{n}(x)\) for \(x=0.400(0.005) 0.600 .\) Plot your results. Note. This illustrates the Gibbs phenomenon of Section \(14.5\) and the danger of trying to calculate with a series expansion in the vicinity of a discontinuity.

Evaluate \(P_{n}^{m}(0)\). $$ \text { ANS. } \begin{aligned} P_{n}^{m}(0)=&\left\\{\begin{array}{ll} (-1)^{(n-m) / 2} & \frac{(n+m) !}{2^{n}((n-m) / 2) !((n+m) / 2 !)}, & n+m \text { even, } \\ 0, & n+m \text { odd. } \end{array}\right.\\\ & \text { Also, } P_{n}^{m}(0)=(-1)^{(n-m) / 2} \frac{(n+m-1) ! !}{(n-m) ! !}, n+m \text { even. } \end{aligned} $$

The electric current density produced by a \(2 p\) electron in a hydrogen atom is $$ \mathrm{J}=\hat{\varphi} \frac{q h}{32 m a_{0}^{5}} e^{-r / a 0} r \sin \theta $$ Using $$ \mathbf{A}\left(\mathbf{r}_{1}\right)=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}_{2}\right)}{\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|} d^{3} r_{2} $$ find the magnetic vector potential produced by this hydrogen electron. Hint. Resolve into Cartesian components. Use the addition theorem to eliminate \(\gamma\), the angle included between \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\).

The hemisphere defined by \(r=a, 0 \leq \theta<\pi / 2\) has an electrostatic potential \(+V_{0}\). The hemisphere \(r=a, \pi / 2<\theta \leq \pi\) has an electrostatic potential \(-V_{0}\). Show that the potential at interior points is $$ \begin{aligned} V &=V_{0} \sum_{n=0}^{\infty} \frac{4 n+3}{2 n+2}\left(\frac{r}{a}\right)^{2 n+1} P_{2 n}(0) P_{2 n+1}(\cos \theta) \\ &=V_{0} \sum_{n=0}^{\infty}(-1)^{n} \frac{(4 n+3)(2 n-1) ! !}{(2 n+2) ! !}\left(\frac{r}{a}\right)^{2 n+1} P_{2 n+1}(\cos \theta) . \end{aligned} $$

Show that each term in the summation $$ \sum_{r=[n / 2]+1}^{n}\left(\frac{d}{d x}\right)^{n} \frac{(-1)^{r} n !}{r !(n-r) !} x^{2 n-2 r} $$ vanishes \((r\) and \(n\) integral).
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