Для вычисления нормы решения уравнения для геометрии необходимо вычислить норму сферической функции Бесселя. Она отличается от обычной функции Бесселя множителем 1 r \frac{\displaystyle 1}{\displaystyle \sqrt r} r 1 ρ ( r ) = r 2 \rho(r) = r^2 ρ ( r ) = r 2
где μ \mu μ d J k + 1 / 2 ( r ) d r ∣ r = μ = 1 2 ⋅ μ ⋅ J k + 1 / 2 ( μ ) \frac{\displaystyle d J_{k + 1/2}(r)}{\displaystyle dr} \bigg|_{r=\mu} = \frac{\displaystyle 1}{\displaystyle 2 \cdot \mu} \cdot J_{k + 1/2}(\mu) d r d J k + 1/2 ( r ) r = μ = 2 ⋅ μ 1 ⋅ J k + 1/2 ( μ ) d d r ( 1 r ⋅ J k + 1 / 2 ( r ) ) ∣ r = μ = 0 \frac{\displaystyle d}{\displaystyle dr} \left( \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot J_{k + 1/2}(r) \right) \bigg|_{r=\mu} = 0 d r d ( r 1 ⋅ J k + 1/2 ( r ) ) r = μ = 0
Попробуем вычислить норму (H.1) ∫ 0 μ ( 1 r ⋅ J k + 1 / 2 ( r ) ) 2 ⋅ r 2 d r = ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r , \int_0^{\mu} \left( \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot J_{k + 1/2}(r) \right)^2 \cdot r^2 \,dr = \int_0^{\mu} r \cdot J_{k + 1/2}^2(r) \,dr, ∫ 0 μ ( r 1 ⋅ J k + 1/2 ( r ) ) 2 ⋅ r 2 d r = ∫ 0 μ r ⋅ J k + 1/2 2 ( r ) d r , C.2 ∫ r m + 1 ⋅ J m ( r ) d r = r m + 1 ⋅ J m + 1 ( r ) + C , \int r^{m+1} \cdot J_m(r) \,dr = r^{m+1} \cdot J_{m+1}(r) + C, ∫ r m + 1 ⋅ J m ( r ) d r = r m + 1 ⋅ J m + 1 ( r ) + C , C.4 r ⋅ d J m ( r ) d r = m ⋅ J m ( r ) − r ⋅ J m + 1 ( r ) , при m = 0 ⇒ d J 0 ( r ) d r = − J 1 ( r ) , r \cdot \frac{\displaystyle d J_m(r)}{\displaystyle dr} = m \cdot J_m(r) - r \cdot J_{m+1}(r), \quad \text{при } m = 0 \;\Rightarrow\; \frac{\displaystyle d J_0(r)}{\displaystyle dr} = - J_1(r), r ⋅ d r d J m ( r ) = m ⋅ J m ( r ) − r ⋅ J m + 1 ( r ) , при m = 0 ⇒ d r d J 0 ( r ) = − J 1 ( r ) ,
∣ u = r − k − 1 / 2 ⋅ J k + 1 / 2 ( r ) , d v = r k + 3 / 2 ⋅ J k + 1 / 2 ( r ) d r d u = ( − k − 1 2 ) ⋅ r − k − 3 / 2 ⋅ J k + 1 / 2 ( r ) d r d u + r − k − 3 / 2 ⋅ [ ( k + 1 2 ) ⋅ J k + 1 / 2 ( r ) − r ⋅ J k + 3 / 2 ( r ) ] d r d u = − r − k − 1 / 2 ⋅ J k + 3 / 2 ( r ) d r v = r k + 3 / 2 ⋅ J k + 3 / 2 ( r ) ∣ \begin{aligned}\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 |\end{aligned} u = r − k − 1/2 ⋅ J k + 1/2 ( r ) , d v = r k + 3/2 ⋅ J k + 1/2 ( r ) d r d u = ( − k − 2 1 ) ⋅ r − k − 3/2 ⋅ J k + 1/2 ( r ) d r d u + r − k − 3/2 ⋅ [ ( k + 2 1 ) ⋅ J k + 1/2 ( r ) − r ⋅ J k + 3/2 ( r ) ] d r d u = − r − k − 1/2 ⋅ J k + 3/2 ( r ) d r v = r k + 3/2 ⋅ J k + 3/2 ( r ) ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r = r ⋅ J k + 1 / 2 ( r ) ⋅ J k + 3 / 2 ( r ) ∣ 0 μ + ∫ 0 μ r ⋅ J k + 3 / 2 2 ( r ) d r . \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr = r \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \bigg|_0^{\mu} + \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr. ∫ 0 μ r ⋅ J k + 1/2 2 ( r ) d r = r ⋅ J k + 1/2 ( r ) ⋅ J k + 3/2 ( r ) 0 μ + ∫ 0 μ r ⋅ J k + 3/2 2 ( r ) d r . Преобразуем уравнение для собственных значений с помощью формулы (C.4 r ⋅ d J m ( r ) d r = m ⋅ J m ( r ) − r ⋅ J m + 1 ( r ) , при m = 0 ⇒ d J 0 ( r ) d r = − J 1 ( r ) , r \cdot \frac{\displaystyle d J_m(r)}{\displaystyle dr} = m \cdot J_m(r) - r \cdot J_{m+1}(r), \quad \text{при } m = 0 \;\Rightarrow\; \frac{\displaystyle d J_0(r)}{\displaystyle dr} = - J_1(r), r ⋅ d r d J m ( r ) = m ⋅ J m ( r ) − r ⋅ J m + 1 ( r ) , при m = 0 ⇒ d r d J 0 ( r ) = − J 1 ( r ) , r = μ r = \mu r = μ
μ ⋅ d J k + 1 / 2 ( r ) d r ∣ r = μ = ( k + 1 2 ) ⋅ J k + 1 / 2 ( μ ) − μ ⋅ J k + 3 / 2 ( μ ) = 1 2 ⋅ J k + 1 / 2 ( μ ) , \mu \cdot \frac{\displaystyle d J_{k + 1/2}(r)}{\displaystyle dr} \bigg|_{r=\mu} = \left( k + \frac{1}{2} \right) \cdot J_{k + 1/2}(\mu) - \mu \cdot J_{k + 3/2}(\mu) = \frac{\displaystyle 1}{\displaystyle 2} \cdot J_{k + 1/2}(\mu), μ ⋅ d r d J k + 1/2 ( r ) r = μ = ( k + 2 1 ) ⋅ J k + 1/2 ( μ ) − μ ⋅ J k + 3/2 ( μ ) = 2 1 ⋅ J k + 1/2 ( μ ) , k ⋅ J k + 1 / 2 ( μ ) = μ ⋅ J k + 3 / 2 ( μ ) , k \cdot J_{k + 1/2}(\mu) = \mu \cdot J_{k + 3/2}(\mu), k ⋅ J k + 1/2 ( μ ) = μ ⋅ J k + 3/2 ( μ ) , и подставим полученное соотношение во внеинтегральный член
∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r = k ⋅ J k + 1 / 2 2 ( μ ) + ∫ 0 μ r ⋅ J k + 3 / 2 2 ( r ) d r . \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr = k \cdot J_{k+1/2}^2(\mu) + \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr. ∫ 0 μ r ⋅ J k + 1/2 2 ( r ) d r = k ⋅ J k + 1/2 2 ( μ ) + ∫ 0 μ r ⋅ J k + 3/2 2 ( r ) d r . Полученный интеграл также возьмём по частям:
∣ u = J k + 3 / 2 2 ( r ) , d u = 2 ⋅ J k + 3 / 2 ( r ) ⋅ 1 r ⋅ ( − ( k + 3 / 2 ) ⋅ J k + 3 / 2 ( r ) + r ⋅ J k + 1 / 2 ( r ) ) d r d v = r d r , v = r 2 / 2 ∣ \begin{aligned}\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 |\end{aligned} u = J k + 3/2 2 ( r ) , d v = r d r , d u = 2 ⋅ J k + 3/2 ( r ) ⋅ r 1 ⋅ ( − ( k + 3/2 ) ⋅ J k + 3/2 ( r ) + r ⋅ J k + 1/2 ( r ) ) d r v = r 2 /2 ∫ 0 μ r ⋅ J k + 3 / 2 2 ( r ) d r = k 2 2 ⋅ J k + 1 / 2 2 ( μ ) + ( k + 3 2 ) ⋅ ∫ 0 μ r ⋅ J k + 3 / 2 2 ( r ) d r − ∫ 0 μ r 2 ⋅ J k + 1 / 2 ( r ) ⋅ J k + 3 / 2 ( r ) d r . \begin{split}
&\int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr = \frac{\displaystyle k^2}{\displaystyle 2} \cdot J_{k+1/2}^2(\mu) + \left( k + \frac{3}{2} \right) \cdot \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr -\\
&\int_0^{\mu} r^2 \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \,dr.
\end{split} ∫ 0 μ r ⋅ J k + 3/2 2 ( r ) d r = 2 k 2 ⋅ J k + 1/2 2 ( μ ) + ( k + 2 3 ) ⋅ ∫ 0 μ r ⋅ J k + 3/2 2 ( r ) d r − ∫ 0 μ r 2 ⋅ J k + 1/2 ( r ) ⋅ J k + 3/2 ( r ) d r . Последний интеграл возьмём по частям:
∣ u = r k + 5 / 2 ⋅ J k + 1 / 2 ( r ) , d v = r − k − 1 / 2 ⋅ J k + 3 / 2 ( r ) d r d u = ( k + 5 2 ) ⋅ r k + 3 / 2 ⋅ J k + 1 / 2 ( r ) + r k + 3 / 2 ⋅ [ ( k + 1 2 ) ⋅ J k + 1 / 2 ( r ) − r ⋅ J k + 3 / 2 ( r ) ] d r d u = ( 2 ⋅ k + 3 ) ⋅ r k + 3 / 2 ⋅ J k + 1 / 2 ( r ) − r k + 5 / 2 ⋅ J k + 3 / 2 ( r ) d r v = − r − k − 1 / 2 ⋅ J k + 1 / 2 ( r ) ∣ \begin{aligned}\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 |\end{aligned} u = r k + 5/2 ⋅ J k + 1/2 ( r ) , d v = r − k − 1/2 ⋅ J k + 3/2 ( r ) d r d u = ( k + 2 5 ) ⋅ r k + 3/2 ⋅ J k + 1/2 ( r ) + r k + 3/2 ⋅ [ ( k + 2 1 ) ⋅ J k + 1/2 ( r ) − r ⋅ J k + 3/2 ( r ) ] d r d u = ( 2 ⋅ k + 3 ) ⋅ r k + 3/2 ⋅ J k + 1/2 ( r ) − r k + 5/2 ⋅ J k + 3/2 ( r ) d r v = − r − k − 1/2 ⋅ J k + 1/2 ( r ) ∫ 0 μ r 2 ⋅ J k + 1 / 2 ( r ) ⋅ J k + 3 / 2 ( r ) d r = − μ 2 ⋅ J k + 1 / 2 2 ( μ ) + ( 2 ⋅ k + 3 ) ⋅ ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r − ∫ 0 μ r 2 ⋅ J k + 1 / 2 ( r ) ⋅ J k + 3 / 2 ( r ) d r , \begin{split}
&\int_0^{\mu} r^2 \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \,dr = - \mu^2 \cdot J_{k+1/2}^2(\mu) +\\
&(2 \cdot k + 3) \cdot \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr - \int_0^{\mu} r^2 \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \,dr,
\end{split} ∫ 0 μ r 2 ⋅ J k + 1/2 ( r ) ⋅ J k + 3/2 ( r ) d r = − μ 2 ⋅ J k + 1/2 2 ( μ ) + ( 2 ⋅ k + 3 ) ⋅ ∫ 0 μ r ⋅ J k + 1/2 2 ( r ) d r − ∫ 0 μ r 2 ⋅ J k + 1/2 ( r ) ⋅ J k + 3/2 ( r ) d r , ∫ 0 μ r 2 ⋅ J k + 1 / 2 ( r ) ⋅ J k + 3 / 2 ( r ) d r = − μ 2 2 ⋅ J k + 1 / 2 2 ( μ ) + ( 2 ⋅ k + 3 ) 2 ⋅ ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r . \int_0^{\mu} r^2 \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \,dr = - \frac{\displaystyle \mu^2}{\displaystyle 2} \cdot J_{k+1/2}^2(\mu) + \frac{\displaystyle (2 \cdot k + 3)}{\displaystyle 2} \cdot \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr. ∫ 0 μ r 2 ⋅ J k + 1/2 ( r ) ⋅ J k + 3/2 ( r ) d r = − 2 μ 2 ⋅ J k + 1/2 2 ( μ ) + 2 ( 2 ⋅ k + 3 ) ⋅ ∫ 0 μ r ⋅ J k + 1/2 2 ( r ) d r . Подставляя найденные равенства последовательно одно в другое, выражаем норму:
0 = k 2 2 ⋅ J k + 1 / 2 2 ( μ ) + ( k + 1 2 ) ⋅ ∫ 0 μ r ⋅ J k + 3 / 2 2 ( r ) d r − ∫ 0 μ r 2 ⋅ J k + 1 / 2 ( r ) ⋅ J k + 3 / 2 ( r ) d r , 0 = \frac{\displaystyle k^2}{\displaystyle 2} \cdot J_{k+1/2}^2(\mu) + \left( k + \frac{1}{2} \right) \cdot \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr - \int_0^{\mu} r^2 \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \,dr, 0 = 2 k 2 ⋅ J k + 1/2 2 ( μ ) + ( k + 2 1 ) ⋅ ∫ 0 μ r ⋅ J k + 3/2 2 ( r ) d r − ∫ 0 μ r 2 ⋅ J k + 1/2 ( r ) ⋅ J k + 3/2 ( r ) d r , ∫ 0 μ r ⋅ J k + 3 / 2 2 ( r ) d r = − k 2 2 ⋅ k + 1 ⋅ J k + 1 / 2 2 ( μ ) + 2 2 ⋅ k + 1 ⋅ ∫ 0 μ r 2 ⋅ J k + 1 / 2 ( r ) ⋅ J k + 3 / 2 ( r ) d r , \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr = - \frac{\displaystyle k^2}{\displaystyle 2 \cdot k + 1} \cdot J_{k+1/2}^2(\mu) + \frac{\displaystyle 2}{\displaystyle 2 \cdot k + 1} \cdot \int_0^{\mu} r^2 \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \,dr, ∫ 0 μ r ⋅ J k + 3/2 2 ( r ) d r = − 2 ⋅ k + 1 k 2 ⋅ J k + 1/2 2 ( μ ) + 2 ⋅ k + 1 2 ⋅ ∫ 0 μ r 2 ⋅ J k + 1/2 ( r ) ⋅ J k + 3/2 ( r ) d r , ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r = k ⋅ ( k + 1 ) 2 ⋅ k + 1 ⋅ J k + 1 / 2 2 ( μ ) + 2 2 ⋅ k + 1 ⋅ ∫ 0 μ r 2 ⋅ J k + 1 / 2 ( r ) ⋅ J k + 3 / 2 ( r ) d r , \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr = \frac{\displaystyle k \cdot (k+1)}{\displaystyle 2 \cdot k + 1} \cdot J_{k+1/2}^2(\mu) + \frac{\displaystyle 2}{\displaystyle 2 \cdot k + 1} \cdot \int_0^{\mu} r^2 \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \,dr, ∫ 0 μ r ⋅ J k + 1/2 2 ( r ) d r = 2 ⋅ k + 1 k ⋅ ( k + 1 ) ⋅ J k + 1/2 2 ( μ ) + 2 ⋅ k + 1 2 ⋅ ∫ 0 μ r 2 ⋅ J k + 1/2 ( r ) ⋅ J k + 3/2 ( r ) d r , ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r = k ⋅ ( k + 1 ) − μ 2 2 ⋅ k + 1 ⋅ J k + 1 / 2 2 ( μ ) + 2 ⋅ k + 3 2 ⋅ k + 1 ⋅ ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r , \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr = \frac{\displaystyle k \cdot (k+1) - \mu^2}{\displaystyle 2 \cdot k + 1} \cdot J_{k+1/2}^2(\mu) + \frac{\displaystyle 2 \cdot k + 3}{\displaystyle 2 \cdot k + 1} \cdot \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr, ∫ 0 μ r ⋅ J k + 1/2 2 ( r ) d r = 2 ⋅ k + 1 k ⋅ ( k + 1 ) − μ 2 ⋅ J k + 1/2 2 ( μ ) + 2 ⋅ k + 1 2 ⋅ k + 3 ⋅ ∫ 0 μ r ⋅ J k + 1/2 2 ( r ) d r , k ⋅ ( k + 1 ) − μ 2 2 ⋅ k + 1 ⋅ J k + 1 / 2 2 ( μ ) = − 2 2 ⋅ k + 1 ⋅ ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r . \frac{\displaystyle k \cdot (k+1) - \mu^2}{\displaystyle 2 \cdot k + 1} \cdot J_{k+1/2}^2(\mu) = - \frac{\displaystyle 2}{\displaystyle 2 \cdot k + 1} \cdot \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr. 2 ⋅ k + 1 k ⋅ ( k + 1 ) − μ 2 ⋅ J k + 1/2 2 ( μ ) = − 2 ⋅ k + 1 2 ⋅ ∫ 0 μ r ⋅ J k + 1/2 2 ( r ) d r . В итоге получаем: