# Bessel equation

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## 11—20 of 101 matching pages

##### 12: 11.10 Anger–Weber Functions
The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation
##### 14: 10.15 Derivatives with Respect to Order
10.15.1 $\frac{\partial J_{\pm\nu}\left(z\right)}{\partial\nu}=\pm J_{\pm\nu}\left(z% \right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}z)^{\pm\nu}\sum_{k=0}^{% \infty}(-1)^{k}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+1\pm\nu\right)}% \frac{(\tfrac{1}{4}z^{2})^{k}}{k!},$
##### 15: 10.45 Functions of Imaginary Order
With $z=x$, and $\nu$ replaced by $i\nu$, the modified Bessel’s equation (10.25.1) becomes …
##### 16: 10.24 Functions of Imaginary Order
With $z=x$ and $\nu$ replaced by $i\nu$, Bessel’s equation (10.2.1) becomes …
##### 17: 12.18 Methods of Computation
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …
##### 18: 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.
• ##### 19: 29.18 Mathematical Applications
(29.18.5) is the differential equation of spherical Bessel functions (§10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1). …
##### 20: 13.6 Relations to Other Functions
13.6.11_2 $M\left(\nu+\tfrac{1}{2},2\nu+1-n,2z\right)=\Gamma\left(\nu-n\right){\mathrm{e}% }^{z}\left(\ifrac{z}{2}\right)^{n-\nu}\sum_{k=0}^{n}(-1)^{k}\frac{{\left(-n% \right)_{k}}{\left(2\nu-2n\right)_{k}}(\nu-n+k)}{{\left(2\nu+1-n\right)_{k}}k!% }I_{\nu+k-n}\left(z\right).$