Incomplete Bessel functions

In mathematics, the Incomplete Bessel functions are types of special functions which act as a type of extension from the complete-type of Bessel functions.

Definition

The Incomplete Bessel functions are defined as the same delay differential equations of the complete-type Bessel functions:

$J_{\alpha -1}(z,w)-J_{\alpha +1}(z,w)=2{\dfrac {\partial }{\partial z}}J_{\alpha }(z,w)$

$Y_{\alpha -1}(z,w)-Y_{\alpha +1}(z,w)=2{\dfrac {\partial }{\partial z}}Y_{\alpha }(z,w)$

$I_{\alpha -1}(z,w)+I_{\alpha +1}(z,w)=2{\dfrac {\partial }{\partial z}}I_{\alpha }(z,w)$

$K_{\alpha -1}(z,w)+K_{\alpha +1}(z,w)=-2{\dfrac {\partial }{\partial z}}K_{\alpha }(z,w)$

External links

• https://www.cambridge.org/core/services/aop-cambridge-core/content/view/9C572E5CE44E9E0DE8630755DF99ABAC/S0013091505000490a.pdf/incomplete-bessel-functions-i.pdf