Bessel equation

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2: 10.1 Special Notation
For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
3: 10.2 Definitions
§10.2(ii) Standard Solutions
The notation $\mathscr{C}_{\nu}\left(z\right)$ denotes $J_{\nu}\left(z\right)$, $Y_{\nu}\left(z\right)$, ${H^{(1)}_{\nu}}\left(z\right)$, ${H^{(2)}_{\nu}}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$.
4: 10.25 Definitions
§10.25(i) Modified Bessel’s Equation
This equation is obtained from Bessel’s equation (10.2.1) on replacing $z$ by $\pm iz$, and it has the same kinds of singularities. …
5: 10.72 Mathematical Applications
§10.72(i) Differential Equations with Turning Points
Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. …
7: 11.9 Lommel Functions
The inhomogeneous Bessel differential equationFor uniform asymptotic expansions, for large $\nu$ and fixed $\mu=-1,0,1,2,\dots$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). … …
10: 28.10 Integral Equations
§28.10(ii) Equations with Bessel-Function Kernels
28.10.9 $\int_{0}^{\ifrac{\pi}{2}}J_{0}\left(2\sqrt{q({\cos}^{2}\tau-{\sin}^{2}\zeta)}% \right)\mathrm{ce}_{2n}\left(\tau,q\right)\mathrm{d}\tau=w_{\mbox{\tiny II}}(% \tfrac{1}{2}\pi;a_{2n}\left(q\right),q)\mathrm{ce}_{2n}\left(\zeta,q\right),$
28.10.10 $\int_{0}^{\pi}J_{0}\left(2\sqrt{q}(\cos\tau+\cos\zeta)\right)\mathrm{ce}_{n}% \left(\tau,q\right)\mathrm{d}\tau=w_{\mbox{\tiny II}}(\pi;a_{n}\left(q\right),% q)\mathrm{ce}_{n}\left(\zeta,q\right).$