球贝塞尔函数
1球贝塞尔方程（spherical Bessel's equation）为
\begin{equation}
x^2 \frac{\mathrm{d}^{2}{y}}{\mathrm{d}{x}^{2}} + 2x \frac{\mathrm{d}{y}}{\mathrm{d}{x}} + [x^2 - n(n + 1)]y = 0
\end{equation}
Fig. 1：球贝塞尔函数和球汉克尔函数（来自 Wikipedia）
两个线性无关的解分别为第一类球贝塞尔函数 $j_n(x)$ 和第二类球贝塞尔函数 $y_n(x)$，见fig. 1 ．它们可以通过贝塞尔函数 $J$，$Y$ 来定义
\begin{equation}
j_n(x) = \sqrt{\frac{\pi}{2x}} J_{n+1/2}(x)
\qquad
y_n(x) = \sqrt{\frac{\pi}{2x}} Y_{n+1/2}(x)
\end{equation}
\begin{equation}
h_n^{(1)}(x) = \sqrt {\frac{\pi }{2x}} H_{n+1/2}^{(1)}(x) = j_n(x) + \mathrm{i} y_n(x)
\end{equation}
\begin{equation}
h_n^{(2)}(x) = \sqrt{\frac{\pi }{2x}} H_{n+1/2}^{(2)}(x) = j_n(x) - \mathrm{i} y_n(x)
\end{equation}
\begin{equation}
j_n(x) = (-x)^n \left(\frac{1}{x} \frac{\mathrm{d}}{\mathrm{d}{x}} \right) ^n \frac{\sin x}{x}
\end{equation}
\begin{equation}
y_n(x) = -(-x)^n \left(\frac{1}{x} \frac{\mathrm{d}}{\mathrm{d}{x}} \right) ^n \frac{\cos x}{x}
\end{equation}
性质
一阶导数（$f$ 是 $j, y, h^{(1)}, h^{(2)}$ 中的任意一种）
\begin{equation}
f'_n(z) = f_{n-1}(z) - \frac{n+1}{z} f_n(z)
\end{equation}
\begin{equation}
j_l(x) \to \sin\left(x - l\pi /2\right) /x
\qquad
y_l(x) \to - \cos\left(x - l\pi /2\right) /x
\end{equation}
\begin{equation}
h_l^{(1)}(x) \to ( - \mathrm{i} )^{l+1} \mathrm{e} ^{ \mathrm{i} x}/x
\qquad
h_l^{(2)}(x) \to \mathrm{i} ^{l + 1} \mathrm{e} ^{- \mathrm{i} x}/x
\end{equation}
\begin{equation}
\begin{aligned}
\int_0^\infty k'j_l(k'r) \cdot kj_l(kr) r^2 \,\mathrm{d}{r} &= \int_0^\infty \sin\left(k'r - l\pi/2\right) \sin\left(kr - l\pi/2\right) \,\mathrm{d}{r} \\
& = \frac{\pi}{2}[\delta(k'-k) - \delta(k'+k)]
\end{aligned}
\end{equation}
修正球贝塞尔函数
修正球贝塞尔方程为
\begin{equation}
x^2 \frac{\mathrm{d}^{2}{y}}{\mathrm{d}{x}^{2}} + 2x \frac{\mathrm{d}{y}}{\mathrm{d}{x}} - [x^2 + n(n + 1)]y = 0
\end{equation}
\begin{equation}
i_n(x) = \sqrt{\frac{\pi }{2x}} I_{n+1/2}(x) = \mathrm{i} ^{-n} j_n( \mathrm{i} x)
\end{equation}
\begin{equation}
k_n(x) = \sqrt{\frac{2}{\pi x}} K_{n+1/2}(x) = \frac{\pi }{2} \mathrm{i} ^{n + 2} h_n^{(1)}( \mathrm{i} x)
\end{equation}
\begin{equation}
i_n(x) \to \frac{ \mathrm{e} ^x}{2x}
\qquad
k_n(x) \to \frac{\pi}{2} \frac{ \mathrm{e} ^{-x}}{x}
\end{equation}