Для вычисления нормы решения уравнения для геометрии необходимо вычислить норму функции Бесселя, которая определяется следующим образом
где μ \mu μ d J m ( r ) d r ∣ r = μ = 0 \frac{\displaystyle d J_m(r)}{\displaystyle dr} \bigg|_{r=\mu} = 0 d r d J m ( r ) r = μ = 0
Из 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 J m ′ ( μ ) = 0 J_m'(\mu) = 0 J m ′ ( μ ) = 0 m ⋅ J m ( μ ) = μ ⋅ J m + 1 ( μ ) m \cdot J_m(\mu) = \mu \cdot J_{m+1}(\mu) m ⋅ J m ( μ ) = μ ⋅ J m + 1 ( μ )
Попробуем вычислить норму (D.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 Граничный член, в отличие от случая Дирихле, не обращается в нуль:
r ⋅ J m ( r ) ⋅ J m + 1 ( r ) ∣ 0 μ = μ ⋅ J m ( μ ) ⋅ J m + 1 ( μ ) = m ⋅ J m 2 ( μ ) . r \cdot J_m(r) \cdot J_{m+1}(r) \bigg|_0^{\mu} = \mu \cdot J_m(\mu) \cdot J_{m+1}(\mu) = m \cdot J_m^2(\mu). r ⋅ J m ( r ) ⋅ J m + 1 ( r ) 0 μ = μ ⋅ J m ( μ ) ⋅ J m + 1 ( μ ) = m ⋅ J m 2 ( μ ) . ∫ 0 μ r ⋅ J m 2 ( r ) d r = m ⋅ J m 2 ( μ ) + ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r . \int_0^{\mu} r \cdot J_m^2(r) \,dr = m \cdot J_m^2(\mu) + \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr. ∫ 0 μ r ⋅ J m 2 ( r ) d r = m ⋅ J m 2 ( μ ) + ∫ 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 = m 2 2 ⋅ J m 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 m^2}{\displaystyle 2} \cdot J_m^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 m 2 ⋅ J m 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 = − μ 2 ⋅ J m 2 ( μ ) , u \cdot v \bigg|_0^{\mu} = - r^2 \cdot J_m^2(r) \bigg|_0^{\mu} = - \mu^2 \cdot J_m^2(\mu), u ⋅ v 0 μ = − r 2 ⋅ J m 2 ( r ) 0 μ = − μ 2 ⋅ J m 2 ( μ ) , 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 = − μ 2 ⋅ J m 2 ( μ ) + 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 = - \mu^2 \cdot J_m^2(\mu) + 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 = − μ 2 ⋅ J m 2 ( μ ) + 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 = − μ 2 2 ⋅ J m 2 ( μ ) + ( 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 = - \frac{\displaystyle \mu^2}{\displaystyle 2} \cdot J_m^2(\mu) + (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 = − 2 μ 2 ⋅ J m 2 ( μ ) + ( m + 1 ) ⋅ ∫ 0 μ r ⋅ J m 2 ( r ) d r . Подставляя найденные равенства последовательно одно в другое, выражаем норму:
∫ 0 μ r ⋅ J m + 1 2 ( r ) d r = 1 m ⋅ ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r − m 2 ⋅ J m 2 ( μ ) , \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr = \frac{\displaystyle 1}{\displaystyle m} \cdot \int_0^{\mu} r^2 \cdot J_m(r) \cdot J_{m+1}(r) \,dr - \frac{\displaystyle m}{\displaystyle 2} \cdot J_m^2(\mu), ∫ 0 μ r ⋅ J m + 1 2 ( r ) d r = m 1 ⋅ ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r − 2 m ⋅ J m 2 ( μ ) , ∫ 0 μ r ⋅ J m 2 ( r ) d r = m ⋅ J m 2 ( μ ) + 1 m ⋅ ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r − m 2 ⋅ J m 2 ( μ ) , \int_0^{\mu} r \cdot J_m^2(r) \,dr = m \cdot J_m^2(\mu) + \frac{\displaystyle 1}{\displaystyle m} \cdot \int_0^{\mu} r^2 \cdot J_m(r) \cdot J_{m+1}(r) \,dr - \frac{\displaystyle m}{\displaystyle 2} \cdot J_m^2(\mu), ∫ 0 μ r ⋅ J m 2 ( r ) d r = m ⋅ J m 2 ( μ ) + m 1 ⋅ ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r − 2 m ⋅ J m 2 ( μ ) , ∫ 0 μ r ⋅ J m 2 ( r ) d r = m 2 ⋅ J m 2 ( μ ) + 1 m ⋅ ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r , \int_0^{\mu} r \cdot J_m^2(r) \,dr = \frac{\displaystyle m}{\displaystyle 2} \cdot J_m^2(\mu) + \frac{\displaystyle 1}{\displaystyle m} \cdot \int_0^{\mu} r^2 \cdot J_m(r) \cdot J_{m+1}(r) \,dr, ∫ 0 μ r ⋅ J m 2 ( r ) d r = 2 m ⋅ J m 2 ( μ ) + m 1 ⋅ ∫ 0 μ r 2 ⋅ J m ( r ) ⋅ J m + 1 ( r ) d r , ∫ 0 μ r ⋅ J m 2 ( r ) d r = m 2 ⋅ J m 2 ( μ ) − μ 2 2 ⋅ m ⋅ J m 2 ( μ ) + m + 1 m ⋅ ∫ 0 μ r ⋅ J m 2 ( r ) d r , \int_0^{\mu} r \cdot J_m^2(r) \,dr = \frac{\displaystyle m}{\displaystyle 2} \cdot J_m^2(\mu) - \frac{\displaystyle \mu^2}{\displaystyle 2 \cdot m} \cdot J_m^2(\mu) + \frac{\displaystyle m+1}{\displaystyle m} \cdot \int_0^{\mu} r \cdot J_m^2(r) \,dr, ∫ 0 μ r ⋅ J m 2 ( r ) d r = 2 m ⋅ J m 2 ( μ ) − 2 ⋅ m μ 2 ⋅ J m 2 ( μ ) + m m + 1 ⋅ ∫ 0 μ r ⋅ J m 2 ( r ) d r , 1 m ⋅ ∫ 0 μ r ⋅ J m 2 ( r ) d r = − m 2 ⋅ J m 2 ( μ ) + μ 2 2 ⋅ m ⋅ J m 2 ( μ ) . \frac{\displaystyle 1}{\displaystyle m} \cdot \int_0^{\mu} r \cdot J_m^2(r) \,dr = - \frac{\displaystyle m}{\displaystyle 2} \cdot J_m^2(\mu) + \frac{\displaystyle \mu^2}{\displaystyle 2 \cdot m} \cdot J_m^2(\mu). m 1 ⋅ ∫ 0 μ r ⋅ J m 2 ( r ) d r = − 2 m ⋅ J m 2 ( μ ) + 2 ⋅ m μ 2 ⋅ J m 2 ( μ ) . В итоге получаем: