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:

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

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

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

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

${\displaystyle H_{v-1}^{(1)}(z,w)-H_{v+1}^{(1)}(z,w)=2{\dfrac {\partial }{\partial z))H_{v}^{(1)}(z,w)}$

${\displaystyle H_{v-1}^{(2)}(z,w)-H_{v+1}^{(2)}(z,w)=2{\dfrac {\partial }{\partial z))H_{v}^{(2)}(z,w)}$

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

${\displaystyle J_{v-1}(z,w)+J_{v+1}(z,w)={\dfrac {2v}{z))J_{v}(z,w)-{\dfrac {2\tanh vw}{z)){\dfrac {\partial }{\partial w))J_{v}(z,w)}$

${\displaystyle Y_{v-1}(z,w)+Y_{v+1}(z,w)={\dfrac {2v}{z))Y_{v}(z,w)-{\dfrac {2\tanh vw}{z)){\dfrac {\partial }{\partial w))Y_{v}(z,w)}$

${\displaystyle I_{v-1}(z,w)-I_{v+1}(z,w)={\dfrac {2v}{z))I_{v}(z,w)-{\dfrac {2\tanh vw}{z)){\dfrac {\partial }{\partial w))I_{v}(z,w)}$

${\displaystyle K_{v-1}(z,w)-K_{v+1}(z,w)=-{\dfrac {2v}{z))K_{v}(z,w)+{\dfrac {2\tanh vw}{z)){\dfrac {\partial }{\partial w))K_{v}(z,w)}$

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

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

Where the new parameter ${\displaystyle w}$ defines the integral bound of the upper-incomplete form and lower-incomplete form of the modified Bessel function of the second kind:^[1]

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

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

Properties

${\displaystyle J_{v}(z,w)=J_{v}(z)+{\dfrac {e^{\frac {v\pi i}{2))J(iz,v,w)-e^{-{\frac {v\pi i}{2))}J(-iz,v,w)}{i\pi ))}$

${\displaystyle Y_{v}(z,w)=Y_{v}(z)+{\dfrac {e^{\frac {v\pi i}{2))J(iz,v,w)+e^{-{\frac {v\pi i}{2))}J(-iz,v,w)}{\pi ))}$

${\displaystyle I_{-v}(z,w)=I_{v}(z,w)}$ for integer ${\displaystyle v}$

${\displaystyle I_{-v}(z,w)-I_{v}(z,w)=I_{-v}(z)-I_{v}(z)-{\dfrac {2\sin v\pi }{\pi ))J(z,v,w)}$

${\displaystyle I_{v}(z,w)=I_{v}(z)+{\dfrac {J(-z,v,w)-e^{-v\pi i}J(z,v,w)}{i\pi ))}$

${\displaystyle I_{v}(z,w)=e^{-{\frac {v\pi i}{2))}J_{v}(iz,w)}$

${\displaystyle K_{-v}(z,w)=K_{v}(z,w)}$

${\displaystyle K_{v}(z,w)={\dfrac {\pi }{2)){\dfrac {I_{-v}(z,w)-I_{v}(z,w)}{\sin v\pi ))}$ for non-integer ${\displaystyle v}$

${\displaystyle H_{v}^{(1)}(z,w)=J_{v}(z,w)+iY_{v}(z,w)}$

${\displaystyle H_{v}^{(2)}(z,w)=J_{v}(z,w)-iY_{v}(z,w)}$

${\displaystyle H_{-v}^{(1)}(z,w)=e^{v\pi i}H_{v}^{(1)}(z,w)}$

${\displaystyle H_{-v}^{(2)}(z,w)=e^{-v\pi i}H_{v}^{(2)}(z,w)}$

${\displaystyle H_{v}^{(1)}(z,w)={\dfrac {J_{-v}(z,w)-e^{-v\pi i}J_{v}(z,w)}{i\sin v\pi ))={\dfrac {Y_{-v}(z,w)-e^{-v\pi i}Y_{v}(z,w)}{\sin v\pi ))}$ for non-integer ${\displaystyle v}$

${\displaystyle H_{v}^{(2)}(z,w)={\dfrac {e^{v\pi i}J_{v}(z,w)-J_{-v}(z,w)}{i\sin v\pi ))={\dfrac {Y_{-v}(z,w)-e^{v\pi i}Y_{v}(z,w)}{\sin v\pi ))}$ for non-integer ${\displaystyle v}$

Differential equations

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

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

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

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

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

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

Integral representations

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

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

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

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

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