Incomplete Bessel functions
# 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:

    $J_{v-1}(z,w)-J_{v+1}(z,w)=2{\dfrac {\partial }{\partial z}}J_{v}(z,w)$
    $Y_{v-1}(z,w)-Y_{v+1}(z,w)=2{\dfrac {\partial }{\partial z}}Y_{v}(z,w)$
    $I_{v-1}(z,w)+I_{v+1}(z,w)=2{\dfrac {\partial }{\partial z}}I_{v}(z,w)$
    $K_{v-1}(z,w)+K_{v+1}(z,w)=-2{\dfrac {\partial }{\partial z}}K_{v}(z,w)$
    $H_{v-1}^{(1)}(z,w)-H_{v+1}^{(1)}(z,w)=2{\dfrac {\partial }{\partial z}}H_{v}^{(1)}(z,w)$
    $H_{v-1}^{(2)}(z,w)-H_{v+1}^{(2)}(z,w)=2{\dfrac {\partial }{\partial z}}H_{v}^{(2)}(z,w)$

And the following suitable extension forms of delay differential equations from that of the complete-type Bessel functions:

    $J_{v-1}(z,w)+J_{v+1}(z,w)={\dfrac {2v}{z}}J_{v}(z,w)-{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}J_{v}(z,w)$
    $Y_{v-1}(z,w)+Y_{v+1}(z,w)={\dfrac {2v}{z}}Y_{v}(z,w)-{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}Y_{v}(z,w)$
    $I_{v-1}(z,w)-I_{v+1}(z,w)={\dfrac {2v}{z}}I_{v}(z,w)-{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}I_{v}(z,w)$
    $K_{v-1}(z,w)-K_{v+1}(z,w)=-{\dfrac {2v}{z}}K_{v}(z,w)+{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}K_{v}(z,w)$
    $H_{v-1}^{(1)}(z,w)+H_{v+1}^{(1)}(z,w)={\dfrac {2v}{z}}H_{v}^{(1)}(z,w)-{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}H_{v}^{(1)}(z,w)$
    $H_{v-1}^{(2)}(z,w)+H_{v+1}^{(2)}(z,w)={\dfrac {2v}{z}}H_{v}^{(2)}(z,w)-{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}H_{v}^{(2)}(z,w)$

Where the new parameter $w$ defines the integral bound of the upper-incomplete form and lower-incomplete form of the modified Bessel function of the second kind:

    $K_{v}(z,w)=\int _{w}^{\infty }e^{-z\cosh t}\cosh vt~dt$
    $J_{v}(z,w)=\int _{0}^{w}e^{-z\cosh t}\cosh vt~dt$

## Properties

    $J_{v}(z,w)=J_{v}(z)+{\dfrac {e^{\frac {v\pi i}{2}}J(iz,v,w)-e^{-{\frac {v\pi i}{2}}}J(-iz,v,w)}{i\pi }}$
    $Y_{v}(z,w)=Y_{v}(z)+{\dfrac {e^{\frac {v\pi i}{2}}J(iz,v,w)+e^{-{\frac {v\pi i}{2}}}J(-iz,v,w)}{\pi }}$
    $I_{-v}(z,w)=I_{v}(z,w)$ for integer $v$
    $I_{-v}(z,w)-I_{v}(z,w)=I_{-v}(z)-I_{v}(z)-{\dfrac {2\sin v\pi }{\pi }}J(z,v,w)$
    $I_{v}(z,w)=I_{v}(z)+{\dfrac {J(-z,v,w)-e^{-v\pi i}J(z,v,w)}{i\pi }}$
    $I_{v}(z,w)=e^{-{\frac {v\pi i}{2}}}J_{v}(iz,w)$
    $K_{-v}(z,w)=K_{v}(z,w)$
    $K_{v}(z,w)={\dfrac {\pi }{2}}{\dfrac {I_{-v}(z,w)-I_{v}(z,w)}{\sin v\pi }}$ for non-integer $v$
    $H_{v}^{(1)}(z,w)=J_{v}(z,w)+iY_{v}(z,w)$
    $H_{v}^{(2)}(z,w)=J_{v}(z,w)-iY_{v}(z,w)$
    $H_{-v}^{(1)}(z,w)=e^{v\pi i}H_{v}^{(1)}(z,w)$
    $H_{-v}^{(2)}(z,w)=e^{-v\pi i}H_{v}^{(2)}(z,w)$
    $H_{v}^{(1)}(z,w)={\dfrac {J_{-v}(z,w)-e^{-v\pi i}J_{v}(z,w)}{i\sin v\pi }}={\dfrac {Y_{-v}(z,w)-e^{-v\pi i}Y_{v}(z,w)}{\sin v\pi }}$ for non-integer $v$
    $H_{v}^{(2)}(z,w)={\dfrac {e^{v\pi i}J_{v}(z,w)-J_{-v}(z,w)}{i\sin v\pi }}={\dfrac {Y_{-v}(z,w)-e^{v\pi i}Y_{v}(z,w)}{\sin v\pi }}$ for non-integer $v$

## Differential equations

$K_{v}(z,w)$ satisfies the inhomogeneous Bessel's differential equation

    $z^{2}{\dfrac {d^{2}y}{dz^{2}}}+z{\dfrac {dy}{dz}}-(x^{2}+v^{2})y=(v\sinh vw+z\cosh vw\sinh w)e^{-z\cosh w}$

Both $J_{v}(z,w)$ , $Y_{v}(z,w)$ , $H_{v}^{(1)}(z,w)$ and $H_{v}^{(2)}(z,w)$ satisfy the partial differential equation

    $z^{2}{\dfrac {\partial ^{2}y}{\partial z^{2}}}+z{\dfrac {\partial y}{\partial z}}+(z^{2}-v^{2})y-{\dfrac {\partial ^{2}y}{\partial w^{2}}}+2v\tanh vw{\dfrac {\partial y}{\partial w}}=0$

Both $I_{v}(z,w)$ and $K_{v}(z,w)$ satisfy the partial differential equation

    $z^{2}{\dfrac {\partial ^{2}y}{\partial z^{2}}}+z{\dfrac {\partial y}{\partial z}}-(z^{2}+v^{2})y-{\dfrac {\partial ^{2}y}{\partial w^{2}}}+2v\tanh vw{\dfrac {\partial y}{\partial w}}=0$

## Integral representations

Base on the preliminary definitions above, one would derive directly the following integral forms of $J_{v}(z,w)$ , $Y_{v}(z,w)$:

    ${\begin{aligned}J_{v}(z,w)&=J_{v}(z)+{\dfrac {1}{\pi i}}\left(\int _{0}^{w}e^{{\frac {v\pi i}{2}}-iz\cosh t}\cosh vt~dt-\int _{0}^{w}e^{iz\cosh t-{\frac {v\pi i}{2}}}\cosh vt~dt\right)\\&=J_{v}(z)+{\dfrac {1}{\pi i}}\left(\int _{0}^{w}\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt-i\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt-\int _{0}^{w}\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt-i\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\right)\\&=J_{v}(z)+{\dfrac {1}{\pi i}}\left(-2i\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\right)\\&=J_{v}(z)-{\dfrac {2}{\pi }}\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\end{aligned}}$
    ${\begin{aligned}Y_{v}(z,w)&=Y_{v}(z)+{\dfrac {1}{\pi }}\left(\int _{0}^{w}e^{{\frac {v\pi i}{2}}-iz\cosh t}\cosh vt~dt+\int _{0}^{w}e^{iz\cosh t-{\frac {v\pi i}{2}}}\cosh vt~dt\right)\\&=Y_{v}(z)+{\dfrac {1}{\pi }}\left(\int _{0}^{w}\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt-i\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt+\int _{0}^{w}\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt+i\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\right)\\&=Y_{v}(z)+{\dfrac {2}{\pi }}\int _{0}^{w}\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\end{aligned}}$

With the Mehler–Sonine integral expressions of $J_{v}(z)={\dfrac {2}{\pi }}\int _{0}^{\infty }\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt$ and $Y_{v}(z)=-{\dfrac {2}{\pi }}\int _{0}^{\infty }\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt$ mentioned in Digital Library of Mathematical Functions,

we can further simplify to $J_{v}(z,w)={\dfrac {2}{\pi }}\int _{w}^{\infty }\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt$ and $Y_{v}(z,w)=-{\dfrac {2}{\pi }}\int _{w}^{\infty }\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt$ , but the issue is not quite good since the convergence range will reduce greatly to $|v|<1$.

