Для вычисления нормы решения уравнения для геометрии необходимо вычислить норму функции Бесселя, которая определяется следующим образом
где μ \mu μ J m ( μ ) = 0 J_m(\mu) = 0 J m ( μ ) = 0
Известны следующие интегралы
а также формулы для производных функции Бесселя
Попробуем вычислить норму (C.1) ∫ 0 μ r ⋅ J m 2 ( r ) d r , \int_0^{\mu} r \cdot J_m^2(r) \,dr, ∫ 0 μ r ⋅ J m 2 ( r ) d r ,
∫ 0 μ r ⋅ J m 2 ( r ) d r = ∣ u = r − m ⋅ J m ( r ) , v = r m + 1 ⋅ J m + 1 ( r ) d u = − m ⋅ r − m − 1 ⋅ J m ( r ) d r + r − m − 1 ⋅ [ m ⋅ J m ( r ) − r ⋅ J m + 1 ( r ) ] d r d v = r m + 1 ⋅ J m ( r ) d r ∣ \int_0^{\mu} r \cdot J_m^2(r) \,dr = \left |
\begin{array}{l}
u = r^{-m} \cdot J_m(r), \quad v = r^{m+1} \cdot J_{m+1}(r)\\
du = -m \cdot r^{-m-1} \cdot J_m(r) \,dr + r^{-m-1} \cdot \left[ m \cdot J_m(r) - r \cdot J_{m+1}(r) \right] \,dr\\
dv = r^{m+1} \cdot J_m(r) \,dr
\end{array}
\right | ∫ 0 μ r ⋅ J m 2 ( r ) d r = u = r − m ⋅ J m ( r ) , v = r m + 1 ⋅ J m + 1 ( r ) d u = − m ⋅ r − m − 1 ⋅ J m ( r ) d r + r − m − 1 ⋅ [ m ⋅ J m ( r ) − r ⋅ J m + 1 ( r ) ] d r d v = r m + 1 ⋅ J m ( r ) d r ∫ 0 μ r ⋅ J m 2 ( r ) d r = r ⋅ J m ( r ) ⋅ J m + 1 ( r ) ∣ 0 μ + ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r = ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r . \int_0^{\mu} r \cdot J_m^2(r) \,dr = r \cdot J_m(r) \cdot J_{m+1}(r) \bigg|_0^{\mu} + \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr = \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr. ∫ 0 μ r ⋅ J m 2 ( r ) d r = r ⋅ J m ( r ) ⋅ J m + 1 ( r ) 0 μ + ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r = ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r . Полученный интеграл также возьмём по частям:
∫ 0 μ r ⋅ J m + 1 2 ( r ) d r = ∣ u = J m + 1 2 ( r ) , d u = 2 ⋅ J m + 1 ( r ) ⋅ 1 r ⋅ [ − ( m + 1 ) ⋅ J m + 1 ( r ) + r ⋅ J m ( r ) ] d r d v = r d r , v = r 2 / 2 ∣ \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr = \left |
\begin{array}{ll}
u = J_{m+1}^2(r), &du = 2 \cdot J_{m+1}(r) \cdot \frac{\displaystyle 1}{\displaystyle r} \cdot \left[ - (m+1) \cdot J_{m+1}(r) + r \cdot J_m(r) \right] \,dr\\
dv = r \,dr, &v = r^2 / 2
\end{array}
\right | ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r = u = J m + 1 2 ( r ) , d v = r d r , d u = 2 ⋅ J m + 1 ( r ) ⋅ r 1 ⋅ [ − ( m + 1 ) ⋅ J m + 1 ( r ) + r ⋅ J m ( r ) ] d r v = r 2 /2 ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r = μ 2 2 ⋅ J m + 1 2 ( μ ) + ( m + 1 ) ⋅ ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r − ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r . \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr = \frac{\displaystyle \mu^2}{\displaystyle 2} \cdot J_{m+1}^2(\mu) + (m+1) \cdot \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr - \int_0^{\mu} r^2 \cdot J_m(r) \cdot J_{m+1}(r) \,dr. ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r = 2 μ 2 ⋅ J m + 1 2 ( μ ) + ( m + 1 ) ⋅ ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r − ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r . Последний интеграл возьмём по частям:
∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r = ∣ u = r m + 2 ⋅ J m ( r ) , v = − r − m ⋅ J m ( r ) d u = ( m + 2 ) ⋅ r m + 1 ⋅ J m ( r ) + r m + 1 ⋅ [ m ⋅ J m ( r ) − r ⋅ J m + 1 ( r ) ] d r d v = r − m ⋅ J m + 1 ( r ) d r ∣ \int_0^{\mu} r^2 \cdot J_m(r) \cdot J_{m+1}(r) \,dr = \left |
\begin{array}{l}
u = r^{m+2} \cdot J_m(r), \quad v = - r^{-m} \cdot J_m(r)\\
du = (m+2) \cdot r^{m+1} \cdot J_m(r) + r^{m+1} \cdot \left[ m \cdot J_m(r) - r \cdot J_{m+1}(r) \right] \,dr\\
dv = r^{-m} \cdot J_{m+1}(r) \,dr
\end{array}
\right | ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r = u = r m + 2 ⋅ J m ( r ) , v = − r − m ⋅ J m ( r ) d u = ( m + 2 ) ⋅ r m + 1 ⋅ J m ( r ) + r m + 1 ⋅ [ m ⋅ J m ( r ) − r ⋅ J m + 1 ( r ) ] d r d v = r − m ⋅ J m + 1 ( r ) d r u ⋅ v ∣ 0 μ = − r 2 ⋅ J m 2 ( r ) ∣ 0 μ = 0 , u \cdot v \bigg|_0^{\mu} = - r^2 \cdot J_m^2(r) \bigg|_0^{\mu} = 0, u ⋅ v 0 μ = − r 2 ⋅ J m 2 ( r ) 0 μ = 0 , d u ⋅ v = − 2 ⋅ ( m + 1 ) ⋅ r ⋅ J m 2 ( r ) + r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) , du \cdot v = - 2 \cdot (m+1) \cdot r \cdot J_m^2(r) + r^2 \cdot J_m(r) \cdot J_{m+1}(r), d u ⋅ v = − 2 ⋅ ( m + 1 ) ⋅ r ⋅ J m 2 ( r ) + r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) , ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r = 2 ⋅ ( m + 1 ) ⋅ ∫ 0 μ r ⋅ J m 2 ( r ) d r − ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r , \int_0^{\mu} r^2 \cdot J_m(r) \cdot J_{m+1}(r) \,dr = 2 \cdot (m+1) \cdot \int_0^{\mu} r \cdot J_m^2(r) \,dr - \int_0^{\mu} r^2 \cdot J_m(r) \cdot J_{m+1}(r) \,dr, ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r = 2 ⋅ ( m + 1 ) ⋅ ∫ 0 μ r ⋅ J m 2 ( r ) d r − ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r , ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r = ( m + 1 ) ⋅ ∫ 0 μ r ⋅ J m 2 ( r ) d r . \int_0^{\mu} r^2 \cdot J_m(r) \cdot J_{m+1}(r) \,dr = (m+1) \cdot \int_0^{\mu} r \cdot J_m^2(r) \,dr. ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r = ( m + 1 ) ⋅ ∫ 0 μ r ⋅ J m 2 ( r ) d r . Подставим последнее равенство в полученное ранее выражение для интеграла ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r ∫ 0 μ r ⋅ J m 2 ( r ) d r = ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r \int_0^{\mu} r \cdot J_m^2(r) \,dr = \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr ∫ 0 μ r ⋅ J m 2 ( r ) d r = ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r ( m + 1 ) (m+1) ( m + 1 )