# 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 equation
11.9.2 $w=s_{{\mu},{\nu}}\left(z\right)+AJ_{\nu}\left(z\right)+BY_{\nu}\left(z\right),$
11.9.7 $s_{{\mu},{\nu}}\left(z\right)=2^{\mu+1}\sum_{k=0}^{\infty}\*\frac{(2k+\mu+1)% \Gamma\left(k+\mu+1\right)}{k!(2k+\mu-\nu+1)(2k+\mu+\nu+1)}J_{2k+\mu+1}\left(z% \right),$
For 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). … …
##### 9: 11.2 Definitions
###### Modified Struve’s Equation
11.2.11 $w=\mathbf{H}_{\nu}\left(x\right)+AJ_{\nu}\left(x\right)+BY_{\nu}\left(x\right),$
11.2.12 $w=\mathbf{K}_{\nu}\left(x\right)+AJ_{\nu}\left(x\right)+BY_{\nu}\left(x\right).$
11.2.16 $w=\mathbf{L}_{\nu}\left(z\right)+AK_{\nu}\left(z\right)+BI_{\nu}\left(z\right),$