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Special Functions

Legendre Polynomials¶

\begin{matrix} (1 - x^{2}) y^{″} - 2 x y^{'} + λ y = 0 \\ (1 - x^{2}) y^{″} - 2 x y^{'} + n (n + 1) y = 0 \\ [(1 - x^{2}) y^{'}] + λ y = 0 \end{matrix}

solve the equation with $y = \sum a_{m} x^{m}$ , we can get the conclusion that

\begin{matrix} λ_{n} = n (n + 1), for n = 0, 1, 2, \dots \\ a_{m + 2} = \frac{m (m + 1) - λ_{n}}{(m + 2) (m + 1)} a_{m} \end{matrix}

thus $a_{n + 2} = a_{n + 4} = \dots = 0$

Normalization¶

Let the polynomial $P_{0} (x) = 1$ , $P_{1} (x) = x$
Recurrence relation : (背)

\begin{array}{r} (n + 1) P_{n + 1} = (2 n + 1) x P_{n} - n P_{n - 1} \end{array}

Fourier-Legendre Expansion¶

$f (x)$ is defined for $- 1 < x < 1$

\begin{matrix} f (x) = \sum_{n = 0}^{\infty} c_{n} P_{n} (x) \\ c_{n} = \frac{(f, P_{n})}{(P_{n}, P_{n})} = \frac{\int_{- 1}^{1} f (x) P_{n} (x) d x}{\int_{- 1}^{1} P_{n}^{2} (x) d x} \\ \int_{- 1}^{1} P_{n}^{2} (x) d x = \frac{2}{2 n + 1} \end{matrix}

Bessel Functions¶

definition

\begin{array}{r} x^{2} y^{″} + x y^{'} + (x^{2} - ν^{2}) y = 0 \end{array}

solution ( for $J_{ν}$ and $J_{- ν}$ are independent)
- $2 ν \notin ℕ$
- $2 ν$ is odd positive integer

\begin{matrix} y (x) = c_{1} J_{ν} (x) + c_{2} J_{- ν} (x) \end{matrix}

solution ( for $J_{ν}$ and $J_{- ν}$ are not independent)
- $2 ν$ is even positive integer

\begin{matrix} y (x) = c_{1} J_{ν} (x) + c_{2} Y_{ν} (x) \end{matrix}

$J_{ν}$ is called a Bessel function of the 1st kind of order $ν$

\begin{matrix} J_{ν} = \sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{2^{n + ν} n! Γ (n + ν + 1)} x^{2 n + ν} \end{matrix}

$Y_{ν} \to Y_{0} (0) = - \infty$

Recurrence Relations¶

\begin{matrix} \frac{d}{d x} (x^{ν} J_{ν}) = x^{ν} J_{ν - 1} \\ \frac{d}{d x} (x^{- ν} J_{ν}) = - x^{- ν} J_{ν + 1} \end{matrix}