# modified Bessel equation

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##### 3: 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. …
##### 4: 10.73 Physical Applications
10.73.3 $\nabla^{4}W+\lambda^{2}\frac{{\partial}^{2}W}{{\partial t}^{2}}=0.$
##### 6: 11.9 Lommel Functions
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). … …
##### 7: 10.45 Functions of Imaginary Order
With $z=x$, and $\nu$ replaced by $i\nu$, the modified Bessel’s equation (10.25.1) becomes …
##### 8: 13.6 Relations to Other Functions
13.6.11_2 $M\left(\nu+\tfrac{1}{2},2\nu+1-n,2z\right)=\Gamma\left(\nu-n\right){\mathrm{e}% }^{z}\left(\ifrac{z}{2}\right)^{n-\nu}\sum_{k=0}^{n}(-1)^{k}\frac{{\left(-n% \right)_{k}}{\left(2\nu-2n\right)_{k}}(\nu-n+k)}{{\left(2\nu+1-n\right)_{k}}k!% }I_{\nu+k-n}\left(z\right).$
##### 9: Bibliography G
• A. Gil, J. Segura, and N. M. Temme (2004b) Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments. ACM Trans. Math. Software 30 (2), pp. 145–158.
• ##### 10: 9.13 Generalized Airy Functions
and $\mathscr{Z}_{p}$ is any linear combination of the modified Bessel functions $I_{p}$ and $e^{p\pi\mathrm{i}}K_{p}$10.25(ii)). Swanson and Headley (1967) define independent solutions $A_{n}\left(z\right)$ and $B_{n}\left(z\right)$ of (9.13.1) by …