User:Egm6321.f09.Team1.sallstrom/HW7

= Problem 1: Even and Oddness of 2nd Solution to Laplace Equation = From lecture note slide 37-1.

Given
The general form of the Legendre polynomials can be written
 * {| style="width:100%" border="0" align="left"

P_n(x) = \sum_{i=0}^{[n/2]} (-1)^i \frac{(2 n - 2 i)! x^{n - 2i}}{2^n i! (n-i)! (n-2i)!} $$ $$ where $$\displaystyle [n/2]$$ means the integer part of $$\displaystyle n/2$$.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }

The second solution to Laplace equation is
 * {| style="width:100%" border="0" align="left"

Q_n(x) = P_n(x) \tanh^{-1}(x) - 2 \sum_{j=1,3,5}^J \frac{2n - 2j + 1}{(2n - j + 1)j} P_{n-j}(x) $$ $$ where $$\displaystyle J := 1 + 2 \left[ \frac{n-1}{2} \right]$$, and the square bracket means the integer of.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }

Find
Use Eq. 2 to show when $$\displaystyle Q_n$$ is even or odd, depending on $$\displaystyle n$$.

Solution
= Problem 2: Plot of Solutions to the Legendre Differential Equation = From lecture note slide 37-1.

Given
The general form of the Legendre polynomials can be written
 * {| style="width:100%" border="0" align="left"

P_n(x) = \sum_{i=0}^{[n/2]} (-1)^i \frac{(2 n - 2 i)! x^{n - 2i}}{2^n i! (n-i)! (n-2i)!} $$ $$ where $$\displaystyle [n/2]$$ means the integer part of $$\displaystyle n/2$$.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }

The second solution to Laplace equation is
 * {| style="width:100%" border="0" align="left"

Q_n(x) = P_n(x) \tanh^{-1}(x) - 2 \sum_{j=1,3,5}^J \frac{2n - 2j + 1}{(2n - j + 1)j} P_{n-j}(x) $$ $$ where $$\displaystyle J := 1 + 2 \left[ \frac{n-1}{2} \right]$$, and the square bracket means the integer of.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }

Find
Plot $$\displaystyle \{P_0, P_1, ..., P_4 \}$$ and $$\displaystyle \{Q_0, Q_1, ..., Q_4 \}$$.

Solution
= Problem 3: Orthogonality of Legendre Functions I = From lecture note slide 37-2.

Given
The general form of the Legendre polynomials can be written
 * {| style="width:100%" border="0" align="left"

P_n(x) = \sum_{i=0}^{[n/2]} (-1)^i \frac{(2 n - 2 i)! x^{n - 2i}}{2^n i! (n-i)! (n-2i)!} $$ $$ where $$\displaystyle [n/2]$$ means the integer part of $$\displaystyle n/2$$.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }

The second solution to Laplace equation is
 * {| style="width:100%" border="0" align="left"

Q_n(x) = P_n(x) \tanh^{-1}(x) - 2 \sum_{j=1,3,5}^J \frac{2n - 2j + 1}{(2n - j + 1)j} P_{n-j}(x) $$ $$ where $$\displaystyle J := 1 + 2 \left[ \frac{n-1}{2} \right]$$, and the square bracket means the integer of.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }

Find
Show that
 * {| style="width:100%" border="0" align="left"

< P_n, Q_n > = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 3)
 * }
 * }

Solution
= Problem 4: Orthogonality of Legendre Functions II = From lecture note slide 37-2.

Given
The following expression is given
 * {| style="width:100%" border="0" align="left"

\int_{-1}^1 L_m \left[ (1-x^2) L'_n \right]' \, {\rm d} x+ n(n+1) \int_{-1}^1 L_m L_n \, {\rm d} x = 0 $$ $$ where $$\displaystyle L_n$$ and $$\displaystyle L_m$$ are solutions to the Legendre equation.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }

Find
Show that, by integrating the first term of Eq. 1, you obtain the expression
 * {| style="width:100%" border="0" align="left"

-\int_{-1}^1 (1-x^2) L'_n L'_m + n (n + 1) = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }

Solution
= Problem 5: = From lecture note slide 38-3.

= Problem 6: = From lecture note slide 38-4.

= Contributing Team Members = Egm6321.f09.Team1.sallstrom 16:56, 18 November 2009 (UTC)