球贝塞尔函数
贡献者： addis
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 - l(l + 1)]y = 0~.
\end{equation}
图 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 $j_l(x)$ 和第二类球贝塞尔函数 $y_l(x)$，见图 1 。它们可以通过贝塞尔函数 $J_l$，$Y_l$ 来定义
\begin{equation}
j_l(x) = \sqrt{\frac{\pi}{2x}} J_{l+1/2}(x)~,
\qquad
y_l(x) = \sqrt{\frac{\pi}{2x}} Y_{l+1/2}(x)~.
\end{equation}
\begin{equation}
h_l^{(1)}(x) = \sqrt {\frac{\pi }{2x}} H_{l+1/2}^{(1)}(x) = j_l(x) + \mathrm{i} y_l(x)~,
\end{equation}
\begin{equation}
h_l^{(2)}(x) = \sqrt{\frac{\pi }{2x}} H_{l+1/2}^{(2)}(x) = j_l(x) - \mathrm{i} y_l(x)~.
\end{equation}
\begin{equation}
j_l(x) = (-x)^l \left(\frac{1}{x} \frac{\mathrm{d}}{\mathrm{d}{x}} \right) ^l \frac{\sin x}{x}~,
\end{equation}
\begin{equation}
y_l(x) = -(-x)^l \left(\frac{1}{x} \frac{\mathrm{d}}{\mathrm{d}{x}} \right) ^l \frac{\cos x}{x}~.
\end{equation}
\begin{align}
&j_0(x) = \frac{\sin x}{x}~,\\
&j_1(x) = \frac{\sin x}{x^2} - \frac{\cos x}{x}~,\\
&j_2(x) = \left(\frac{3}{x^2} - 1 \right) \frac{\sin x}{x} - \frac{3\cos x}{x^2}~,\\
&j_3(x) = \left(\frac{15}{x^3} - \frac{6}{x} \right) \frac{\sin x}{x} - \left(\frac{15}{x^2}-1 \right) \frac{\cos x}{x}~.
\end{align}
\begin{align}
&y_0(x) = -\frac{\cos x}{x}~,\\
&y_1(x) = -\frac{\cos x}{x^2} - \frac{\sin x}{x}~,\\
&y_2(x) = \left(-\frac{3}{x^2}+1 \right) \frac{\cos x}{x} - \frac{3\sin x}{x^2}~,\\
&y_3(x) = \left(-\frac{15}{x^3}+\frac{6}{x} \right) \frac{\cos x}{x} - \left(\frac{15}{x^2} - 1 \right) \frac{\sin x}{x}~.
\end{align}
l = 4; Series[SphericalBesselJ[l, x], {x, \[Infinity], 1000}] //
Normal // Simplify
1. 性质
原点值
\begin{equation}
j_l(0) = \delta_{l,0} ~,\qquad y_l(0) = -\infty~.
\end{equation}
奇偶性：当 $l$ 为偶数，$j_l, y_l$ 分别是偶函数和奇函数，而 $l$ 为奇数时则相反。
\begin{equation}
j_l(-x) = (-1)^l j_l(x)~,
\qquad
y_l(-y) = -(-1)^l y_l(x)~.
\end{equation}
一阶导数满足（$f$ 是 $j, y, h^{(1)}, h^{(2)}$ 中的任意一种）
\begin{equation}
f'_l(z) = f_{l-1}(z) - \frac{l+1}{z} f_l(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}
j_l(\rho) = \frac{(- \mathrm{i} )^l}{2} \int_{-1}^1 P_l(x) \mathrm{e} ^{ \mathrm{i} \rho x} \,\mathrm{d}{x} ~.
\end{equation}
\begin{equation}
j_l(x_1 \pm x_2) = \sum_{l_1,l_2} \mathrm{i} ^{l_1\pm l_2-l} \frac{(2l_1+1)(2l_2+1)}{2l+1}
\begin{bmatrix}l_1 & l_2 & l\\ 0 & 0 & 0\end{bmatrix} ^2 j_{l_1}(x_1) j_{l_2}(x_2)~.
\end{equation}
2. 积分性质
由渐进形式可得径向归一化积分为（$\delta$ 是狄拉克 $\delta$ 函数，推导见式 20 ）
\begin{equation}
\begin{aligned}
\int_0^\infty k'r j_l(k'r) \cdot kr j_l(kr) \,\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) \qquad (k, k' > 0)~,
\end{aligned}
\end{equation}
\begin{equation}
\int_{0}^{\infty} j_l(x) \,\mathrm{d}{x} = \frac{\pi l!}{2^{l+1} [(l/2)!]^2}~.
\end{equation}
\begin{equation}
\int_{-\infty}^{+\infty} j_l(x) \,\mathrm{d}{x} = \left\{\begin{aligned}
&\frac{\pi l!}{2^l [(l/2)!]^2} &\quad (l = \text{偶数})\\
&0 &\quad (l = \text{奇数})
\end{aligned}\right. ~.\end{equation}
\begin{equation}
\int_{-\infty}^{+\infty} j_l(x) j_{l'}(x) \,\mathrm{d}{x} = \frac{\pi}{2l+1}\delta_{l,l'}~,
\end{equation}
\begin{equation}
\int_0^\infty j_l(k' r) j_l(kr) \,\mathrm{d}{r} = \frac{\pi}{2(2l+1)} \frac{k_<^l}{k_>^{l+1}}~.
\end{equation}
\begin{equation}
\int_{-\infty}^{+\infty} \mathrm{e} ^{ \mathrm{i} k x} j_l(x) \,\mathrm{d}{x} = \pi \mathrm{i} ^l P_l(k)~.
\end{equation}
3. 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 + l(l + 1)]y = 0~.
\end{equation}
\begin{equation}
i_l(x) = \sqrt{\frac{\pi }{2x}} I_{l+1/2}(x) = \mathrm{i} ^{-l} j_l( \mathrm{i} x)~,
\end{equation}
\begin{equation}
k_l(x) = \sqrt{\frac{2}{\pi x}} K_{l+1/2}(x) = \frac{\pi }{2} \mathrm{i} ^{l + 2} h_l^{(1)}( \mathrm{i} x)~,
\end{equation}
\begin{equation}
i_l(x) \to \frac{ \mathrm{e} ^x}{2x}~,
\qquad
k_l(x) \to \frac{\pi}{2} \frac{ \mathrm{e} ^{-x}}{x}~.
\end{equation}
1. ^ 参考 Wikipedia 相关页面。以及 R. Mehrem, arXiv:0909.0494 (2009)。
2. ^ 笔者也不明白为什么要这么做。