Incomplete Bessel Function

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:
:$J_(z,w)-J_(z,w)=2\dfracJ_v(z,w)$
:$Y_(z,w)-Y_(z,w)=2\dfracY_v(z,w)$
:$I_(z,w)+I_(z,w)=2\dfracI_v(z,w)$
:$K_(z,w)+K_(z,w)=-2\dfracK_v(z,w)$
:$H_^(z,w)-H_^(z,w)=2\dfracH_v^(z,w)$
:$H_^(z,w)-H_^(z,w)=2\dfracH_v^(z,w)$
And the following suitable extension forms of delay differential equations from that of the complete-type Bessel functions:
:$J_(z,w)+J_(z,w)=\dfracJ_v(z,w)-\dfrac\dfracJ_v(z,w)$
:$Y_(z,w)+Y_(z,w)=\dfracY_v(z,w)-\dfrac\dfracY_v(z,w)$
:$I_(z,w)-I_(z,w)=\dfracI_v(z,w)-\dfrac\dfracI_v(z,w)$
:$K_(z,w)-K_(z,w)=-\dfracK_v(z,w)+\dfrac\dfracK_v(z,w)$
:$H_^(z,w)+H_^(z,w)=\dfracH_v^(z,w)-\dfrac\dfracH_v^(z,w)$
:$H_^(z,w)+H_^(z,w)=\dfracH_v^(z,w)-\dfrac\dfracH_v^(z,w)$
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:
:$K_v(z,w)=\int_w^\infty e^\cosh vt~dt$
:$J_v(z,w)=\int_0^we^\cosh vt~dt$

Properties

:$J_v(z,w)=J_v(z)+\dfrac$
:$Y_v(z,w)=Y_v(z)+\dfrac$
:$I_(z,w)=I_v(z,w)$ for integer $v$
:$I_(z,w)-I_v(z,w)=I_(z)-I_v(z)-\dfracJ(z,v,w)$
:$I_v(z,w)=I_v(z)+\dfrac$
:$I_v(z,w)=e^J_v(iz,w)$
:$K_(z,w)=K_v(z,w)$
:$K_v(z,w)=\dfrac\dfrac$ for non-integer $v$
:$H_v^(z,w)=J_v(z,w)+iY_v(z,w)$
:$H_v^(z,w)=J_v(z,w)-iY_v(z,w)$
:$H_^(z,w)=e^H_v^(z,w)$
:$H_^(z,w)=e^H_v^(z,w)$
:$H_v^(z,w)=\dfrac=\dfrac$ for non-integer $v$
:$H_v^(z,w)=\dfrac=\dfrac$ for non-integer $v$

Differential equations

$K_v(z,w)$ satisfies the inhomogeneous Bessel's differential equation
:$z^2\dfrac+z\dfrac-(x^2+v^2)y=(v\sinh vw+z\cosh vw\sinh w)e^$
Both $J_v(z,w)$ , $Y_v(z,w)$ , $H_v^(z,w)$ and $H_v^(z,w)$ satisfy the partial differential equation
:$z^2\dfrac+z\dfrac+(z^2-v^2)y-\dfrac+2v\tanh vw\dfrac=0$
Both $I_v(z,w)$ and $K_v(z,w)$ satisfy the partial differential equation
:$z^2\dfrac+z\dfrac-(z^2+v^2)y-\dfrac+2v\tanh vw\dfrac=0$

Integral representations

Base on the preliminary definitions above, one would derive directly the following integral forms of $J_v(z,w)$ , $Y_v(z,w)$:
:$\begin J_v(z,w)&=J_v(z)+\dfrac\left(\int_0^we^\cosh vt~dt-\int_0^we^\cosh vt~dt\right) \\&=J_v(z)+\dfrac\left(\int_0^w\cos\left(z\cosh t-\dfrac\right)\cosh vt~dt-i\int_0^w\sin\left(z\cosh t-\dfrac\right)\cosh vt~dt\right.\\ &\quad\quad\quad\quad\quad\quad\left.-\int_0^w\cos\left(z\cosh t-\dfrac\right)\cosh vt~dt-i\int_0^w\sin\left(z\cosh t-\dfrac\right)\cosh vt~dt\right) \\&=J_v(z)+\dfrac\left(-2i\int_0^w\sin\left(z\cosh t-\dfrac\right)\cosh vt~dt\right) \\&=J_v(z)-\dfrac\int_0^w\sin\left(z\cosh t-\dfrac\right)\cosh vt~dt\end$
:$\begin Y_v(z,w)&=Y_v(z)+\dfrac\left(\int_0^we^\cosh vt~dt+\int_0^we^\cosh vt~dt\right) \\&=Y_v(z)+\dfrac\left(\int_0^w\cos\left(z\cosh t-\dfrac\right)\cosh vt~dt-i\int_0^w\sin\left(z\cosh t-\dfrac\right)\cosh vt~dt\right.\\ &\quad\quad\quad\quad\quad\quad\left.+\int_0^w\cos\left(z\cosh t-\dfrac\right)\cosh vt~dt+i\int_0^w\sin\left(z\cosh t-\dfrac\right)\cosh vt~dt\right) \\&=Y_v(z)+\dfrac\int_0^w\cos\left(z\cosh t-\dfrac\right)\cosh vt~dt\end$
With the Mehler–Sonine integral expressions of $J_v(z)=\dfrac\int_0^\infty\sin\left(z\cosh t-\dfrac\right)\cosh vt~dt$ and $Y_v(z)=-\dfrac\int_0^\infty\cos\left(z\cosh t-\dfrac\right)\cosh vt~dt$ mentioned in Digital Library of Mathematical Functions,

we can further simplify to $J_v(z,w)=\dfrac\int_w^\infty\sin\left(z\cosh t-\dfrac\right)\cosh vt~dt$ and $Y_v(z,w)=-\dfrac\int_w^\infty\cos\left(z\cosh t-\dfrac\right)\cosh vt~dt$ , but the issue is not quite good since the convergence range will reduce greatly to $, v, <1$.

References

External links

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*{{cite journal , last1=Jones , first1=D. S. , title=Incomplete Bessel functions. II. Asymptotic expansions for large argument , journal=Proceedings of the Edinburgh Mathematical Society , date=October 2007 , volume=50 , issue=3 , pages=711–723 , doi=10.1017/S0013091505000908, doi-access=free

Special hypergeometric functions