H. Норма сферической функции Бесселя (Нейман)

Для вычисления нормы решения уравнения для геометрии необходимо вычислить норму сферической функции Бесселя. Она отличается от обычной функции Бесселя множителем 1r\frac{\displaystyle 1}{\displaystyle \sqrt r}, а вес для сферических координат ρ(r)\rho(r)​=r2{} = r^2, поэтому мы приходим к тому же интегралу, что и для обычной функции Бесселя

∫0μ(1r⋅Jk+1/2(r))2⋅r2 dr\displaystyle \int_0^{\mu} \left( \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot J_{k + 1/2}(r) \right)^2 \cdot r^2 \,dr =∫0μr⋅Jk+1/22(r) dr,\displaystyle {} = \int_0^{\mu} r \cdot J_{k + 1/2}^2(r) \,dr,
(H.1)

где μ\mu — одно из решений уравнения dJk+1/2(r)dr∣r=μ\frac{\displaystyle d J_{k + 1/2}(r)}{\displaystyle dr} \bigg|_{r=\mu}​=12⋅μ⋅Jk+1/2(μ){} = \frac{\displaystyle 1}{\displaystyle 2 \cdot \mu} \cdot J_{k + 1/2}(\mu), которое получается из условия Неймана для сферической функции Бесселя: ddr(1r⋅Jk+1/2(r))∣r=μ\frac{\displaystyle d}{\displaystyle dr} \left( \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot J_{k + 1/2}(r) \right) \bigg|_{r=\mu}​=0{} = 0.

Попробуем вычислить норму (H.1) интегрированием по частям, учитывая формулы (C.2) и (C.4):

∣u=r−k−1/2⋅Jk+1/2(r),dv=rk+3/2⋅Jk+1/2(r) drdu=(−k−12)⋅r−k−3/2⋅Jk+1/2(r) drdu+r−k−3/2⋅[(k+12)⋅Jk+1/2(r)−r⋅Jk+3/2(r)] drdu=−r−k−1/2⋅Jk+3/2(r) drv=rk+3/2⋅Jk+3/2(r)∣\left | \begin{aligned} &u = r^{-k - 1/2} \cdot J_{k + 1/2}(r), \quad dv = r^{k+3/2} \cdot J_{k + 1/2}(r) \,dr\\ &du = \left( - k - \frac{1}{2} \right) \cdot r^{-k - 3/2} \cdot J_{k + 1/2}(r) \,dr\\ &\phantom{du} + r^{-k - 3/2} \cdot \left[ \left( k + \frac{1}{2} \right) \cdot J_{k + 1/2}(r) - r \cdot J_{k + 3/2}(r) \right] \,dr\\ &du = - r^{-k - 1/2} \cdot J_{k + 3/2}(r) \,dr\\ &v = r^{k+3/2} \cdot J_{k + 3/2}(r) \end{aligned} \right |
∫0μr⋅Jk+1/22(r) dr\displaystyle \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr =r\displaystyle {} = r​⋅Jk+1/2(r)\displaystyle {} \cdot J_{k+1/2}(r)​⋅Jk+3/2(r)∣0μ\displaystyle {} \cdot J_{k+3/2}(r) \bigg|_0^{\mu}​+∫0μr⋅Jk+3/22(r) dr.\displaystyle {} + \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr.

Преобразуем уравнение для собственных значений с помощью формулы (C.4) при rr​=μ{} = \mu

μ⋅dJk+1/2(r)dr∣r=μ\displaystyle \mu \cdot \frac{\displaystyle d J_{k + 1/2}(r)}{\displaystyle dr} \bigg|_{r=\mu} =(k+12)⋅Jk+1/2(μ)\displaystyle {} = \left( k + \frac{1}{2} \right) \cdot J_{k + 1/2}(\mu)​−μ⋅Jk+3/2(μ)\displaystyle {} - \mu \cdot J_{k + 3/2}(\mu)​=12⋅Jk+1/2(μ),\displaystyle {} = \frac{\displaystyle 1}{\displaystyle 2} \cdot J_{k + 1/2}(\mu),
k⋅Jk+1/2(μ)\displaystyle k \cdot J_{k + 1/2}(\mu) =μ⋅Jk+3/2(μ),\displaystyle {} = \mu \cdot J_{k + 3/2}(\mu),

и подставим полученное соотношение во внеинтегральный член

∫0μr⋅Jk+1/22(r) dr\displaystyle \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr =k⋅Jk+1/22(μ)\displaystyle {} = k \cdot J_{k+1/2}^2(\mu)​+∫0μr⋅Jk+3/22(r) dr.\displaystyle {} + \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr.

Полученный интеграл также возьмём по частям:

∣u=Jk+3/22(r),du=2⋅Jk+3/2(r)⋅1r⋅(−(k+3/2)⋅Jk+3/2(r)+r⋅Jk+1/2(r)) drdv=r dr,v=r2/2∣\left | \begin{array}{ll} u = J_{k+3/2}^2(r), &du = 2 \cdot J_{k+3/2}(r) \cdot \frac{\displaystyle 1}{\displaystyle r} \cdot \left( - (k+3/2) \cdot J_{k+3/2}(r) + r \cdot J_{k+1/2}(r) \right) \,dr\\ dv = r \,dr, &v = r^2 / 2 \end{array} \right |
∫0μr⋅Jk+3/22(r) dr\displaystyle \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr =k22⋅Jk+1/22(μ)\displaystyle {} = \frac{\displaystyle k^2}{\displaystyle 2} \cdot J_{k+1/2}^2(\mu)​+(k+32)\displaystyle {} + \left( k + \frac{3}{2} \right)​⋅∫0μr\displaystyle {} \cdot \int_0^{\mu} r​⋅Jk+3/22(r) dr\displaystyle {} \cdot J_{k+3/2}^2(r) \,dr​−∫0μr2\displaystyle {} - \int_0^{\mu} r^2​⋅Jk+1/2(r)\displaystyle {} \cdot J_{k+1/2}(r)​⋅Jk+3/2(r) dr.\displaystyle {} \cdot J_{k+3/2}(r) \,dr.

Последний интеграл возьмём по частям:

∣u=rk+5/2⋅Jk+1/2(r),dv=r−k−1/2⋅Jk+3/2(r) drdu=(k+52)⋅rk+3/2⋅Jk+1/2(r)+rk+3/2⋅[(k+12)⋅Jk+1/2(r)−r⋅Jk+3/2(r)] drdu=(2⋅k+3)⋅rk+3/2⋅Jk+1/2(r)−rk+5/2⋅Jk+3/2(r) drv=−r−k−1/2⋅Jk+1/2(r)∣\left | \begin{aligned} &u = r^{k+5/2} \cdot J_{k+1/2}(r), \quad dv = r^{-k-1/2} \cdot J_{k+3/2}(r) \,dr\\ &du = \left( k + \frac{5}{2} \right) \cdot r^{k+3/2} \cdot J_{k+1/2}(r) + r^{k+3/2} \cdot \left[ \left( k + \frac{1}{2} \right) \cdot J_{k+1/2}(r) - r \cdot J_{k+3/2}(r) \right] \,dr\\ &du = (2 \cdot k + 3) \cdot r^{k+3/2} \cdot J_{k+1/2}(r) - r^{k+5/2} \cdot J_{k+3/2}(r) \,dr\\ &v = - r^{-k-1/2} \cdot J_{k+1/2}(r) \end{aligned} \right |
∫0μr2⋅Jk+1/2(r)⋅Jk+3/2(r) dr\displaystyle \int_0^{\mu} r^2 \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \,dr =−μ2⋅Jk+1/22(μ)\displaystyle {} = - \mu^2 \cdot J_{k+1/2}^2(\mu)​+(2⋅k+3)\displaystyle {} + (2 \cdot k + 3)​⋅∫0μr\displaystyle {} \cdot \int_0^{\mu} r​⋅Jk+1/22(r) dr\displaystyle {} \cdot J_{k+1/2}^2(r) \,dr​−∫0μr2\displaystyle {} - \int_0^{\mu} r^2​⋅Jk+1/2(r)\displaystyle {} \cdot J_{k+1/2}(r)​⋅Jk+3/2(r) dr,\displaystyle {} \cdot J_{k+3/2}(r) \,dr,
∫0μr2⋅Jk+1/2(r)⋅Jk+3/2(r) dr\displaystyle \int_0^{\mu} r^2 \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \,dr =−μ22⋅Jk+1/22(μ)\displaystyle {} = - \frac{\displaystyle \mu^2}{\displaystyle 2} \cdot J_{k+1/2}^2(\mu)​+(2⋅k+3)2\displaystyle {} + \frac{\displaystyle (2 \cdot k + 3)}{\displaystyle 2}​⋅∫0μr\displaystyle {} \cdot \int_0^{\mu} r​⋅Jk+1/22(r) dr.\displaystyle {} \cdot J_{k+1/2}^2(r) \,dr.

Подставляя найденные равенства последовательно одно в другое, выражаем норму:

0\displaystyle 0 =k22⋅Jk+1/22(μ)\displaystyle {} = \frac{\displaystyle k^2}{\displaystyle 2} \cdot J_{k+1/2}^2(\mu)​+(k+12)\displaystyle {} + \left( k + \frac{1}{2} \right)​⋅∫0μr\displaystyle {} \cdot \int_0^{\mu} r​⋅Jk+3/22(r) dr\displaystyle {} \cdot J_{k+3/2}^2(r) \,dr​−∫0μr2\displaystyle {} - \int_0^{\mu} r^2​⋅Jk+1/2(r)\displaystyle {} \cdot J_{k+1/2}(r)​⋅Jk+3/2(r) dr,\displaystyle {} \cdot J_{k+3/2}(r) \,dr,
∫0μr⋅Jk+3/22(r) dr\displaystyle \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr =−k22⋅k+1⋅Jk+1/22(μ)\displaystyle {} = - \frac{\displaystyle k^2}{\displaystyle 2 \cdot k + 1} \cdot J_{k+1/2}^2(\mu)​+22⋅k+1\displaystyle {} + \frac{\displaystyle 2}{\displaystyle 2 \cdot k + 1}​⋅∫0μr2\displaystyle {} \cdot \int_0^{\mu} r^2​⋅Jk+1/2(r)\displaystyle {} \cdot J_{k+1/2}(r)​⋅Jk+3/2(r) dr,\displaystyle {} \cdot J_{k+3/2}(r) \,dr,
∫0μr⋅Jk+1/22(r) dr\displaystyle \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr =k⋅(k+1)2⋅k+1\displaystyle {} = \frac{\displaystyle k \cdot (k+1)}{\displaystyle 2 \cdot k + 1}​⋅Jk+1/22(μ)\displaystyle {} \cdot J_{k+1/2}^2(\mu)​+22⋅k+1\displaystyle {} + \frac{\displaystyle 2}{\displaystyle 2 \cdot k + 1}​⋅∫0μr2\displaystyle {} \cdot \int_0^{\mu} r^2​⋅Jk+1/2(r)\displaystyle {} \cdot J_{k+1/2}(r)​⋅Jk+3/2(r) dr,\displaystyle {} \cdot J_{k+3/2}(r) \,dr,
∫0μr⋅Jk+1/22(r) dr\displaystyle \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr =k⋅(k+1)−μ22⋅k+1\displaystyle {} = \frac{\displaystyle k \cdot (k+1) - \mu^2}{\displaystyle 2 \cdot k + 1}​⋅Jk+1/22(μ)\displaystyle {} \cdot J_{k+1/2}^2(\mu)​+2⋅k+32⋅k+1\displaystyle {} + \frac{\displaystyle 2 \cdot k + 3}{\displaystyle 2 \cdot k + 1}​⋅∫0μr\displaystyle {} \cdot \int_0^{\mu} r​⋅Jk+1/22(r) dr,\displaystyle {} \cdot J_{k+1/2}^2(r) \,dr,
k⋅(k+1)−μ22⋅k+1⋅Jk+1/22(μ)\displaystyle \frac{\displaystyle k \cdot (k+1) - \mu^2}{\displaystyle 2 \cdot k + 1} \cdot J_{k+1/2}^2(\mu) =−22⋅k+1\displaystyle {} = - \frac{\displaystyle 2}{\displaystyle 2 \cdot k + 1}​⋅∫0μr\displaystyle {} \cdot \int_0^{\mu} r​⋅Jk+1/22(r) dr.\displaystyle {} \cdot J_{k+1/2}^2(r) \,dr.

В итоге получаем:

∫0μr⋅Jk+1/22(r) dr\displaystyle \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr =μ2−k⋅(k+1)2\displaystyle {} = \frac{\displaystyle \mu^2 - k \cdot (k+1)}{\displaystyle 2}​⋅Jk+1/22(μ).\displaystyle {} \cdot J_{k+1/2}^2(\mu).
(H.2)