Math Bits - Legendre's Equation

Students should be able to:

• Write a series as an infinite sum.
• Change the index on an infinite sum.
• Add infinite sums in sum notation.
• Write out the first few terms of an infinite sum.
• Identify that the coefficients of two infinite sums are independently equal if the infinite sums are equal.

1. (SumShiftP)

   In each of the following sums, shift the index $n\rightarrow n+2$. Don't forget to shift the limits of the sum as well. Then write out all of the terms in the sum (if the sum has a finite number of terms) or the first five terms in the sum (if the sum has an infinite number of terms) and convince yourself that the two different expressions for each sum are the same:

   1. $$\sum_{n=0}^3 n$$

   2. $$\sum_{n=1}^5 e^{in\phi}$$

   3. $$\sum_{n=0}^{\infty} a_n n(n-1)z^{n-2}$$

2. (RadialSeries) In this problem students solve the differential equation for the Laguerre Polynomials. The question is worded so that students see this as a practice problem on series solution methods rather than specifically the Laguerre equation which they can look up. By timing the assignment of this homework problem appropriately, it is possible to arrange to have the students complete this problem just before beginning the radial solution to the hydrogen atom; thus saving time in class solving the Laguerre equation.

   1. By hand, find the recurrence relation for a power series solution $H(\rho)$ of the equation:

      $$\rho {d^2 H\over d\rho^2} +(2\ell+2-\rho){dH\over d\rho} +(\lambda-\ell-1) H=0$$

      where $\ell$ is a known positive integer, and $\lambda$ is an unknown constant.

   2. Suppose that you want a solution to (a) which is a polynomial of degree 4. Assume that $\ell=2$. What does that tell you about the unknown constant $\lambda$?

   3. Find the polynomial of degree 4 solution to the differential equation in part (a) assuming $\ell=2$. Assume anything you need to about $\lambda$.

3. (Laguerre)

   Laguerre Polynomials\hfill\break The differential equation for Laguerre polynomials $L_m(z)$ is given by $$zL^{\prime\prime}+(1-z)L^{\prime}+nL=0$$ Find a polynomial solution of this differential equation for the case $n=4$. For what values of $z$ is your solution valid?

1. (Legendre's Equation)

   (Optional if not done in class)

   1. Use a power series expanded about $z = 0$ to find the first six terms in each of two independent solutions to this differential equation for $l = 2$.

   2. For what values of $z$ do you expect your power series solutions to converge?

   3. Find at least one solution to this differential equation for $l = 2$ that does converge outside the range you identified above.

2. (LegendrePoly)

   1. Use Mathematica or Maple to find the first 5 Legendre polynomials.

   2. Use Rodrigues' formula to calculate the first 5 Legendre polynomials. (You are encouraged to use Mathematica or Maple to help with the derivatives.

   3. Look up two recurrence relations for Legendre polynomials and use them to find $P_3(z)$ and $P'_3(z)$ assuming that all you know is that $P_0(z)=1$ and $P_1(z)=2z$. Do this part of the problem by hand.

3. (LaplaceSeparate)