C. Bessel function norm (Dirichlet)

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,
(C.1)

where μ\mu is one of the solutions of the equation Jm(μ)J_m(\mu)​=0{} = 0.

The following integrals are known

∫rm+1⋅Jm(r) dr\displaystyle \int r^{m+1} \cdot J_m(r) \,dr =rm+1⋅Jm+1(r)\displaystyle {} = r^{m+1} \cdot J_{m+1}(r)​+C,\displaystyle {} + C,
(C.2)
∫r−m⋅Jm+1(r) dr\displaystyle \int r^{-m} \cdot J_{m+1}(r) \,dr =−r−m⋅Jm(r)\displaystyle {} = - r^{-m} \cdot J_m(r)​+C,\displaystyle {} + C,
(C.3)

as well as the formulas for the derivatives of the Bessel function

r⋅dJm(r)dr\displaystyle r \cdot \frac{\displaystyle d J_m(r)}{\displaystyle dr} =m⋅Jm(r)\displaystyle {} = m \cdot J_m(r)​−r⋅Jm+1(r),\displaystyle {} - r \cdot J_{m+1}(r),​as m\displaystyle \text{as } m​=0  \displaystyle {} = 0 \;​⇒  dJ0(r)dr\displaystyle {} \Rightarrow\; \frac{\displaystyle d J_0(r)}{\displaystyle dr}​=−J1(r),\displaystyle {} = - J_1(r),
(C.4)
r⋅dJm(r)dr\displaystyle r \cdot \frac{\displaystyle d J_m(r)}{\displaystyle dr} =−m⋅Jm(r)\displaystyle {} = - m \cdot J_m(r)​+r⋅Jm−1(r),\displaystyle {} + r \cdot J_{m-1}(r),​as m\displaystyle \text{as } m​=0  \displaystyle {} = 0 \;​⇒  dJ0(r)dr\displaystyle {} \Rightarrow\; \frac{\displaystyle d J_0(r)}{\displaystyle dr}​=J−1(r).\displaystyle {} = J_{-1}(r).
(C.5)

Let us try to compute the norm (C.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 |
∫0μr⋅Jm2(r) dr\displaystyle \int_0^{\mu} r \cdot J_m^2(r) \,dr =r⋅Jm(r)⋅Jm+1(r)∣0μ\displaystyle {} = r \cdot J_m(r) \cdot J_{m+1}(r) \bigg|_0^{\mu}​+∫0μr⋅Jm+12(r) dr\displaystyle {} + \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr​=∫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 =μ22⋅Jm+12(μ)\displaystyle {} = \frac{\displaystyle \mu^2}{\displaystyle 2} \cdot J_{m+1}^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}​=0,\displaystyle {} = 0,
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 =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 =(m+1)⋅∫0μr⋅Jm2(r) dr.\displaystyle {} = (m+1) \cdot \int_0^{\mu} r \cdot J_m^2(r) \,dr.

Let us substitute the last equality into the previously obtained expression for the integral ∫0μr⋅Jm+12(r) dr\int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr. Taking into account the equality ∫0μr⋅Jm2(r) dr\int_0^{\mu} r \cdot J_m^2(r) \,dr​=∫0μr⋅Jm+12(r) dr{} = \int_0^{\mu} r \cdot J_{m+1}^2(r) \,dr, the terms with (m+1)(m+1) cancel, and we finally obtain:

∫0μr⋅Jm2(r) dr\displaystyle \int_0^{\mu} r \cdot J_m^2(r) \,dr =μ22⋅Jm+12(μ).\displaystyle {} = \frac{\displaystyle \mu^2}{\displaystyle 2} \cdot J_{m+1}^2(\mu).
(C.6)