# 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 $\int_{0}^{\pi}J_{0}\left(2\sqrt{q}(\cos\tau+\cos\zeta)\right)\mathrm{ce}_{n}% \left(\tau,q\right)\mathrm{d}\tau=w_{\mbox{\tiny II}}(\pi;a_{n}\left(q\right),% q)\mathrm{ce}_{n}\left(\zeta,q\right).$