Incomplete Bessel functions

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

:$J\_\{v-1\}(z,w)-J\_\{v+1\}(z,w)=2\backslash dfrac\{\backslash partial\}\{\backslash partial\; z\}J\_v(z,w)$

:$Y\_\{v-1\}(z,w)-Y\_\{v+1\}(z,w)=2\backslash dfrac\{\backslash partial\}\{\backslash partial\; z\}Y\_v(z,w)$

:$I\_\{v-1\}(z,w)+I\_\{v+1\}(z,w)=2\backslash dfrac\{\backslash partial\}\{\backslash partial\; z\}I\_v(z,w)$

:$K\_\{v-1\}(z,w)+K\_\{v+1\}(z,w)=-2\backslash dfrac\{\backslash partial\}\{\backslash partial\; z\}K\_v(z,w)$

:$H\_\{v-1\}^\{(1)\}(z,w)-H\_\{v+1\}^\{(1)\}(z,w)=2\backslash dfrac\{\backslash partial\}\{\backslash partial\; z\}H\_v^\{(1)\}(z,w)$

:$H\_\{v-1\}^\{(2)\}(z,w)-H\_\{v+1\}^\{(2)\}(z,w)=2\backslash dfrac\{\backslash partial\}\{\backslash partial\; z\}H\_v^\{(2)\}(z,w)$

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

:$J\_\{v-1\}(z,w)+J\_\{v+1\}(z,w)=\backslash dfrac\{2v\}\{z\}J\_v(z,w)-\backslash dfrac\{2\backslash tanh\; vw\}\{z\}\backslash dfrac\{\backslash partial\}\{\backslash partial\; w\}J\_v(z,w)$

:$Y\_\{v-1\}(z,w)+Y\_\{v+1\}(z,w)=\backslash dfrac\{2v\}\{z\}Y\_v(z,w)-\backslash dfrac\{2\backslash tanh\; vw\}\{z\}\backslash dfrac\{\backslash partial\}\{\backslash partial\; w\}Y\_v(z,w)$

:$I\_\{v-1\}(z,w)-I\_\{v+1\}(z,w)=\backslash dfrac\{2v\}\{z\}I\_v(z,w)-\backslash dfrac\{2\backslash tanh\; vw\}\{z\}\backslash dfrac\{\backslash partial\}\{\backslash partial\; w\}I\_v(z,w)$

:$K\_\{v-1\}(z,w)-K\_\{v+1\}(z,w)=-\backslash dfrac\{2v\}\{z\}K\_v(z,w)+\backslash dfrac\{2\backslash tanh\; vw\}\{z\}\backslash dfrac\{\backslash partial\}\{\backslash partial\; w\}K\_v(z,w)$

:$H\_\{v-1\}^\{(1)\}(z,w)+H\_\{v+1\}^\{(1)\}(z,w)=\backslash dfrac\{2v\}\{z\}H\_v^\{(1)\}(z,w)-\backslash dfrac\{2\backslash tanh\; vw\}\{z\}\backslash dfrac\{\backslash partial\}\{\backslash partial\; w\}H\_v^\{(1)\}(z,w)$

:$H\_\{v-1\}^\{(2)\}(z,w)+H\_\{v+1\}^\{(2)\}(z,w)=\backslash dfrac\{2v\}\{z\}H\_v^\{(2)\}(z,w)-\backslash dfrac\{2\backslash tanh\; vw\}\{z\}\backslash dfrac\{\backslash partial\}\{\backslash partial\; w\}H\_v^\{(2)\}(z,w)$

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:{{cite journal |last1=Jones |first1=D. S. |title=Incomplete Bessel functions. I |journal=Proceedings of the Edinburgh Mathematical Society |date=February 2007 |volume=50 |issue=1 |pages=173–183 |doi=10.1017/S0013091505000490 |doi-access = free }}

:$K\_v(z,w)=\backslash int\_w^\backslash infty\; e^\{-z\backslash cosh\; t\}\backslash cosh\; vt~dt$

:$J\_v(z,w)=\backslash int\_0^we^\{-z\backslash cosh\; t\}\backslash cosh\; vt~dt$

:$J\_v(z,w)=J\_v(z)+\backslash dfrac\{e^\backslash frac\{v\backslash pi\; i\}\{2\}J(iz,v,w)-e^\{-\backslash frac\{v\backslash pi\; i\}\{2\}\}J(-iz,v,w)\}\{i\backslash pi\}$

:$Y\_v(z,w)=Y\_v(z)+\backslash dfrac\{e^\backslash frac\{v\backslash pi\; i\}\{2\}J(iz,v,w)+e^\{-\backslash frac\{v\backslash pi\; i\}\{2\}\}J(-iz,v,w)\}\{\backslash pi\}$

:$I\_\{-v\}(z,w)=I\_v(z,w)$ for integer $v$

:$I\_\{-v\}(z,w)-I\_v(z,w)=I\_\{-v\}(z)-I\_v(z)-\backslash dfrac\{2\backslash sin\; v\backslash pi\}\{\backslash pi\}J(z,v,w)$

:$I\_v(z,w)=I\_v(z)+\backslash dfrac\{J(-z,v,w)-e^\{-v\backslash pi\; i\}J(z,v,w)\}\{i\backslash pi\}$

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

:$K\_\{-v\}(z,w)=K\_v(z,w)$

:$K\_v(z,w)=\backslash dfrac\{\backslash pi\}\{2\}\backslash dfrac\{I\_\{-v\}(z,w)-I\_v(z,w)\}\{\backslash sin\; v\backslash pi\}$ for non-integer $v$

:$H\_v^\{(1)\}(z,w)=J\_v(z,w)+iY\_v(z,w)$

:$H\_v^\{(2)\}(z,w)=J\_v(z,w)-iY\_v(z,w)$

:$H\_\{-v\}^\{(1)\}(z,w)=e^\{v\backslash pi\; i\}H\_v^\{(1)\}(z,w)$

:$H\_\{-v\}^\{(2)\}(z,w)=e^\{-v\backslash pi\; i\}H\_v^\{(2)\}(z,w)$

:$H\_v^\{(1)\}(z,w)=\backslash dfrac\{J\_\{-v\}(z,w)-e^\{-v\backslash pi\; i\}J\_v(z,w)\}\{i\backslash sin\; v\backslash pi\}=\backslash dfrac\{Y\_\{-v\}(z,w)-e^\{-v\backslash pi\; i\}Y\_v(z,w)\}\{\backslash sin\; v\backslash pi\}$ for non-integer $v$

:$H\_v^\{(2)\}(z,w)=\backslash dfrac\{e^\{v\backslash pi\; i\}J\_v(z,w)-J\_\{-v\}(z,w)\}\{i\backslash sin\; v\backslash pi\}=\backslash dfrac\{Y\_\{-v\}(z,w)-e^\{v\backslash pi\; i\}Y\_v(z,w)\}\{\backslash sin\; v\backslash pi\}$ for non-integer $v$

$K\_v(z,w)$ satisfies the inhomogeneous Bessel's differential equation

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

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

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

Both $I\_v(z,w)$ and $K\_v(z,w)$ satisfy the partial differential equation

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

Base on the preliminary definitions above, one would derive directly the following integral forms of $J\_v(z,w)$ , $Y\_v(z,w)$:

:$\backslash begin\{align\}$

J_v(z,w)&=J_v(z)+\dfrac{1}{\pi i}\left(\int_0^we^{\frac{v\pi i}{2}-iz\cosh t}\cosh vt~dt-\int_0^we^{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\right.\\

&\quad\quad\quad\quad\quad\quad\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\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{align}

:$\backslash begin\{align\}$

Y_v(z,w)&=Y_v(z)+\dfrac{1}{\pi}\left(\int_0^we^{\frac{v\pi i}{2}-iz\cosh t}\cosh vt~dt+\int_0^we^{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\right.\\

&\quad\quad\quad\quad\quad\quad\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\right)

\\&=Y_v(z)+\dfrac{2}{\pi}\int_0^w\cos\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt\end{align}

With the Mehler–Sonine integral expressions of $J\_v(z)=\backslash dfrac\{2\}\{\backslash pi\}\backslash int\_0^\backslash infty\backslash sin\backslash left(z\backslash cosh\; t-\backslash dfrac\{v\backslash pi\}\{2\}\backslash right)\backslash cosh\; vt~dt$ and $Y\_v(z)=-\backslash dfrac\{2\}\{\backslash pi\}\backslash int\_0^\backslash infty\backslash cos\backslash left(z\backslash cosh\; t-\backslash dfrac\{v\backslash pi\}\{2\}\backslash right)\backslash cosh\; vt~dt$ mentioned in Digital Library of Mathematical Functions,{{dlmf|id=10.9.8|title=Bessel Functions|first=R. B. |last=Paris}}

we can further simplify to $J\_v(z,w)=\backslash dfrac\{2\}\{\backslash pi\}\backslash int\_w^\backslash infty\backslash sin\backslash left(z\backslash cosh\; t-\backslash dfrac\{v\backslash pi\}\{2\}\backslash right)\backslash cosh\; vt~dt$ and $Y\_v(z,w)=-\backslash dfrac\{2\}\{\backslash pi\}\backslash int\_w^\backslash infty\backslash cos\backslash left(z\backslash cosh\; t-\backslash dfrac\{v\backslash pi\}\{2\}\backslash right)\backslash cosh\; vt~dt$ , but the issue is not quite good since the convergence range will reduce greatly to .

{{Reflist}}

• {{cite book |last1=Agrest |first1=Matest M. |last2=Maksimov |first2=Michail S. |title=Theory of Incomplete Cylindrical Functions and their Applications |date=1971 |publisher=Springer-Verlag Berlin Heidelberg |location=Berlin, Heidelberg |isbn=978-3-642-65023-9}}
• {{cite journal |last1=Cicchetti |first1=R. |last2=Faraone |first2=A. |title=Incomplete Hankel and Modified Bessel Functions: A Class of Special Functions for Electromagnetics |journal=IEEE Transactions on Antennas and Propagation |date=December 2004 |volume=52 |issue=12 |pages=3373–3389 |doi=10.1109/TAP.2004.835269|bibcode=2004ITAP...52.3373C |s2cid=25089438 }}
• {{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 }}

Category:Special hypergeometric functions