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:

$J_{v-1}(z,w)-J_{v+1}(z,w)=2\frac{\partial }{\partial z}{J}_v(z,w)$

$Y_{v-1}(z,w)-Y_{v+1}(z,w)=2\frac{\partial }{\partial z}{Y}_v(z,w)$

$I_{v-1}(z,w)+I_{v+1}(z,w)=2\frac{\partial }{\partial z}{I}_v(z,w)$

$K_{v-1}(z,w)+K_{v+1}(z,w)=-2\frac{\partial }{\partial z}{K}_v(z,w)$

$H_{v-1}^{(1)}(z,w)-H_{v+1}^{(1)}(z,w)=2\frac{\partial }{\partial z}{H}_v^{(1)}(z,w)$

$H_{v-1}^{(2)}(z,w)-H_{v+1}^{(2)}(z,w)=2\frac{\partial }{\partial z}{H}_v^{(2)}(z,w)$

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

$J_{v-1}(z,w)+J_{v+1}(z,w)=\frac{2v}{z}{J}_v(z,w)-\frac{2\tanh vw}{z}\frac{\partial }{\partial w}{J}_v(z,w)$

$Y_{v-1}(z,w)+Y_{v+1}(z,w)=\frac{2v}{z}{Y}_v(z,w)-\frac{2\tanh vw}{z}\frac{\partial }{\partial w}{Y}_v(z,w)$

$I_{v-1}(z,w)-I_{v+1}(z,w)=\frac{2v}{z}{I}_v(z,w)-\frac{2\tanh vw}{z}\frac{\partial }{\partial w}{I}_v(z,w)$

$K_{v-1}(z,w)-K_{v+1}(z,w)=-\frac{2v}{z}{K}_v(z,w)+\frac{2\tanh vw}{z}\frac{\partial }{\partial w}{K}_v(z,w)$

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

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

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:

$K_v(z,w)=\underset{w}{\overset{\mathrm{\infty }}{\int }}{e}^{-z\cosh t}\cosh vtdt$

$J_v(z,w)=\underset{0}{\overset{w}{\int }}{e}^{-z\cosh t}\cosh vtdt$

Properties

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

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

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

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

$I_v(z,w)=I_v(z)+\frac{J(-z,v,w)-e^{-v\pi i}J(z,v,w)}{i\pi }$

$I_v(z,w)=e^{-\frac{v\pi i}{2}}{J}_v(iz,w)$

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

$K_v(z,w)=\frac{\pi }{2}\frac{I_{-v}(z,w)-I_v(z,w)}{\sin v\pi }$ for non-integer $v$

$H_v^{(1)}(z,w)=J_v(z,w)+i{Y}_v(z,w)$

$H_v^{(2)}(z,w)=J_v(z,w)-i{Y}_v(z,w)$

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

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

$H_v^{(1)}(z,w)=\frac{J_{-v}(z,w)-e^{-v\pi i}{J}_v(z,w)}{i\sin v\pi }=\frac{Y_{-v}(z,w)-e^{-v\pi i}{Y}_v(z,w)}{\sin v\pi }$ for non-integer $v$

$H_v^{(2)}(z,w)=\frac{e^{v\pi i}{J}_v(z,w)-J_{-v}(z,w)}{i\sin v\pi }=\frac{Y_{-v}(z,w)-e^{v\pi i}{Y}_v(z,w)}{\sin v\pi }$ for non-integer $v$

Differential equations

$K_v(z,w)$ satisfies the inhomogeneous Bessel's differential equation

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

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

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

Both $I_v(z,w)$ and $K_v(z,w)$ satisfy the partial differential equation

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

Integral representations

Base on the preliminary definitions above, one would derive directly the following integral forms of $J_v(z,w)$ , $Y_v(z,w)$:

$\begin{array}{cc}{J}_v(z,w)& =J_v(z)+\frac{1}{\pi i}(\underset{0}{\overset{w}{\int }}{e}^{\frac{v\pi i}{2}-iz\cosh t}\cosh vtdt-\underset{0}{\overset{w}{\int }}{e}^{iz\cosh t-\frac{v\pi i}{2}}\cosh vtdt)\\ & =J_v(z)+\frac{1}{\pi i}(\underset{0}{\overset{w}{\int }}\cos (z\cosh t-\frac{v\pi }{2})\cosh vtdt-i\underset{0}{\overset{w}{\int }}\sin (z\cosh t-\frac{v\pi }{2})\cosh vtdt\\ & -\underset{0}{\overset{w}{\int }}\cos (z\cosh t-\frac{v\pi }{2})\cosh vtdt-i\underset{0}{\overset{w}{\int }}\sin (z\cosh t-\frac{v\pi }{2})\cosh vtdt)\\ & =J_v(z)+\frac{1}{\pi i}(-2i\underset{0}{\overset{w}{\int }}\sin (z\cosh t-\frac{v\pi }{2})\cosh vtdt)\\ & =J_v(z)-\frac{2}{\pi }\underset{0}{\overset{w}{\int }}\sin (z\cosh t-\frac{v\pi }{2})\cosh vtdt\end{array}$

$\begin{array}{cc}{Y}_v(z,w)& =Y_v(z)+\frac{1}{\pi }(\underset{0}{\overset{w}{\int }}{e}^{\frac{v\pi i}{2}-iz\cosh t}\cosh vtdt+\underset{0}{\overset{w}{\int }}{e}^{iz\cosh t-\frac{v\pi i}{2}}\cosh vtdt)\\ & =Y_v(z)+\frac{1}{\pi }(\underset{0}{\overset{w}{\int }}\cos (z\cosh t-\frac{v\pi }{2})\cosh vtdt-i\underset{0}{\overset{w}{\int }}\sin (z\cosh t-\frac{v\pi }{2})\cosh vtdt\\ & +\underset{0}{\overset{w}{\int }}\cos (z\cosh t-\frac{v\pi }{2})\cosh vtdt+i\underset{0}{\overset{w}{\int }}\sin (z\cosh t-\frac{v\pi }{2})\cosh vtdt)\\ & =Y_v(z)+\frac{2}{\pi }\underset{0}{\overset{w}{\int }}\cos (z\cosh t-\frac{v\pi }{2})\cosh vtdt\end{array}$

With the Mehler–Sonine integral expressions of $J_v(z)=\frac{2}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}\sin (z\cosh t-\frac{v\pi }{2})\cosh vtdt$ and $Y_v(z)=-\frac{2}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}\cos (z\cosh t-\frac{v\pi }{2})\cosh vtdt$ mentioned in Digital Library of Mathematical Functions,

we can further simplify to $J_v(z,w)=\frac{2}{\pi }\underset{w}{\overset{\mathrm{\infty }}{\int }}\sin (z\cosh t-\frac{v\pi }{2})\cosh vtdt$ and $Y_v(z,w)=-\frac{2}{\pi }\underset{w}{\overset{\mathrm{\infty }}{\int }}\cos (z\cosh t-\frac{v\pi }{2})\cosh vtdt$ , but the issue is not quite good since the convergence range will reduce greatly to $|v|<1$.

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