D. Bessel function norm (Neumann)

To compute the norm of the solution of the equation for the geometry, we need to compute the norm of the Bessel function, which is defined as follows

∫0μr⋅Jm2(r) dr,\int_0^{\mu} r \cdot J_m^2(r) \,dr,
(D.1)

where μ\mu is one of the solutions of the equation dJm(r)dr∣r=μ\frac{\displaystyle d J_m(r)}{\displaystyle dr} \bigg|_{r=\mu}​=0{} = 0.

From formula (C.4) for rr​=μ{} = \mu, where Jm′(μ)J_m'(\mu)​=0{} = 0, we obtain the relation m⋅Jm(μ)m \cdot J_m(\mu)​=μ⋅Jm+1(μ){} = \mu \cdot J_{m+1}(\mu), which is used below.

Let us try to compute the norm (D.1) by integration by parts:

∫0μr⋅Jm2(r) dr\displaystyle \int_0^{\mu} r \cdot J_m^2(r) \,dr =∣u=r−m⋅Jm(r),v=rm+1⋅Jm+1(r)du=−m⋅r−m−1⋅Jm(r) dr+r−m−1⋅[m⋅Jm(r)−r⋅Jm+1(r)] drdv=rm+1⋅Jm(r) dr∣\displaystyle {} = \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 |

The boundary term, unlike the Dirichlet case, does not vanish:

r⋅Jm(r)⋅Jm+1(r)∣0μ\displaystyle r \cdot J_m(r) \cdot J_{m+1}(r) \bigg|_0^{\mu} =μ⋅Jm(μ)⋅Jm+1(μ)\displaystyle {} = \mu \cdot J_m(\mu) \cdot J_{m+1}(\mu)​=m⋅Jm2(μ).\displaystyle {} = m \cdot J_m^2(\mu).
∫0μr⋅Jm2(r) dr\displaystyle \int_0^{\mu} r \cdot J_m^2(r) \,dr =m⋅Jm2(μ)\displaystyle {} = m \cdot J_m^2(\mu)​+∫0μr⋅Jm+12(r) dr.\displaystyle {} + \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr.

We integrate the resulting integral by parts as well:

∫0μr⋅Jm+12(r) dr\displaystyle \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr =∣u=Jm+12(r),du=2⋅Jm+1(r)⋅1r⋅[−(m+1)⋅Jm+1(r)+r⋅Jm(r)] drdv=r dr,v=r2/2∣\displaystyle {} = \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 |
∫0μr⋅Jm+12(r) dr\displaystyle \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr =m22⋅Jm2(μ)\displaystyle {} = \frac{\displaystyle m^2}{\displaystyle 2} \cdot J_m^2(\mu)​+(m+1)⋅∫0μr⋅Jm+12(r) dr\displaystyle {} + (m+1) \cdot \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr​−∫0μr2\displaystyle {} - \int_0^{\mu} r^2​⋅Jm(r)\displaystyle {} \cdot J_m(r)​⋅Jm+1(r) dr.\displaystyle {} \cdot J_{m+1}(r) \,dr.

We integrate the last integral by parts:

∫0μr2⋅Jm(r)⋅Jm+1(r) dr\displaystyle \int_0^{\mu} r^2 \cdot J_m(r) \cdot J_{m+1}(r) \,dr =∣u=rm+2⋅Jm(r),v=−r−m⋅Jm(r)du=(m+2)⋅rm+1⋅Jm(r)+rm+1⋅[m⋅Jm(r)−r⋅Jm+1(r)] drdv=r−m⋅Jm+1(r) dr∣\displaystyle {} = \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 |
u⋅v∣0μ\displaystyle u \cdot v \bigg|_0^{\mu} =−r2⋅Jm2(r)∣0μ\displaystyle {} = - r^2 \cdot J_m^2(r) \bigg|_0^{\mu}​=−μ2⋅Jm2(μ),\displaystyle {} = - \mu^2 \cdot J_m^2(\mu),
du⋅v\displaystyle du \cdot v =−2⋅(m+1)⋅r⋅Jm2(r)\displaystyle {} = - 2 \cdot (m+1) \cdot r \cdot J_m^2(r)​+r2⋅Jm(r)⋅Jm+1(r),\displaystyle {} + r^2 \cdot J_m(r) \cdot J_{m+1}(r),
∫0μr2⋅Jm(r)⋅Jm+1(r) dr\displaystyle \int_0^{\mu} r^2 \cdot J_m(r) \cdot J_{m+1}(r) \,dr =−μ2⋅Jm2(μ)\displaystyle {} = - \mu^2 \cdot J_m^2(\mu)​+2⋅(m+1)⋅∫0μr⋅Jm2(r) dr\displaystyle {} + 2 \cdot (m+1) \cdot \int_0^{\mu} r \cdot J_m^2(r) \,dr​−∫0μr2\displaystyle {} - \int_0^{\mu} r^2​⋅Jm(r)\displaystyle {} \cdot J_m(r)​⋅Jm+1(r) dr,\displaystyle {} \cdot J_{m+1}(r) \,dr,
∫0μr2⋅Jm(r)⋅Jm+1(r) dr\displaystyle \int_0^{\mu} r^2 \cdot J_m(r) \cdot J_{m+1}(r) \,dr =−μ22⋅Jm2(μ)\displaystyle {} = - \frac{\displaystyle \mu^2}{\displaystyle 2} \cdot J_m^2(\mu)​+(m+1)⋅∫0μr⋅Jm2(r) dr.\displaystyle {} + (m+1) \cdot \int_0^{\mu} r \cdot J_m^2(r) \,dr.

Substituting the obtained equalities successively into one another, we express the norm:

∫0μr⋅Jm+12(r) dr\displaystyle \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr =1m\displaystyle {} = \frac{\displaystyle 1}{\displaystyle m}​⋅∫0μr2\displaystyle {} \cdot \int_0^{\mu} r^2​⋅Jm(r)\displaystyle {} \cdot J_m(r)​⋅Jm+1(r) dr\displaystyle {} \cdot J_{m+1}(r) \,dr​−m2⋅Jm2(μ),\displaystyle {} - \frac{\displaystyle m}{\displaystyle 2} \cdot J_m^2(\mu),
∫0μr⋅Jm2(r) dr\displaystyle \int_0^{\mu} r \cdot J_m^2(r) \,dr =m⋅Jm2(μ)\displaystyle {} = m \cdot J_m^2(\mu)​+1m\displaystyle {} + \frac{\displaystyle 1}{\displaystyle m}​⋅∫0μr2\displaystyle {} \cdot \int_0^{\mu} r^2​⋅Jm(r)\displaystyle {} \cdot J_m(r)​⋅Jm+1(r) dr\displaystyle {} \cdot J_{m+1}(r) \,dr​−m2⋅Jm2(μ),\displaystyle {} - \frac{\displaystyle m}{\displaystyle 2} \cdot J_m^2(\mu),
∫0μr⋅Jm2(r) dr\displaystyle \int_0^{\mu} r \cdot J_m^2(r) \,dr =m2⋅Jm2(μ)\displaystyle {} = \frac{\displaystyle m}{\displaystyle 2} \cdot J_m^2(\mu)​+1m\displaystyle {} + \frac{\displaystyle 1}{\displaystyle m}​⋅∫0μr2\displaystyle {} \cdot \int_0^{\mu} r^2​⋅Jm(r)\displaystyle {} \cdot J_m(r)​⋅Jm+1(r) dr,\displaystyle {} \cdot J_{m+1}(r) \,dr,
∫0μr⋅Jm2(r) dr\displaystyle \int_0^{\mu} r \cdot J_m^2(r) \,dr =m2⋅Jm2(μ)\displaystyle {} = \frac{\displaystyle m}{\displaystyle 2} \cdot J_m^2(\mu)​−μ22⋅m⋅Jm2(μ)\displaystyle {} - \frac{\displaystyle \mu^2}{\displaystyle 2 \cdot m} \cdot J_m^2(\mu)​+m+1m⋅∫0μr⋅Jm2(r) dr,\displaystyle {} + \frac{\displaystyle m+1}{\displaystyle m} \cdot \int_0^{\mu} r \cdot J_m^2(r) \,dr,
1m⋅∫0μr⋅Jm2(r) dr\displaystyle \frac{\displaystyle 1}{\displaystyle m} \cdot \int_0^{\mu} r \cdot J_m^2(r) \,dr =−m2⋅Jm2(μ)\displaystyle {} = - \frac{\displaystyle m}{\displaystyle 2} \cdot J_m^2(\mu)​+μ22⋅m⋅Jm2(μ).\displaystyle {} + \frac{\displaystyle \mu^2}{\displaystyle 2 \cdot m} \cdot J_m^2(\mu).

Finally we obtain:

∫0μr⋅Jm2(r) dr\displaystyle \int_0^{\mu} r \cdot J_m^2(r) \,dr =(μ2−m2)⋅Jm2(μ)2.\displaystyle {} = \left( \mu^2 - m^2 \right) \cdot \frac{\displaystyle J_m^2(\mu)}{\displaystyle 2}.
(D.2)