Incomplete Bessel functions explained

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 ^{( 1 )} _{v -1} (z,w) - H ^{( 1 )} _{v +1} (z,w) =2 \dfrac{\partial}{\partial z}H ^{( 1 )} _{v} (z,w) H ^{( 2 )} _{v -1} (z,w) - H ^{( 2 )} _{v +1} (z,w) =2 \dfrac{\partial}{\partial z}H ^{( 2 )} _{v} (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{ 2 v}{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{ 2 v}{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{ 2 v}{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{ 2 v}{z}K _{v(z,w)+\dfrac{2\tanh} vw}{z}\dfrac{\partial}{\partial w}K _{v(z,w)} H ^{( 1 )} _{v -1} (z,w) + H ^{( 1 )} _{v +1} (z,w) = \dfrac{ 2 v}{z}H ^{( 1 )} _{v} (z,w) - \dfrac{ 2 \tanh vw}{z}\dfrac{\partial}{\partial w}H ^{( 1 )} _{v} (z,w) H ^{( 2 )} _{v -1} (z,w) + H ^{( 2 )} _{v +1} (z,w) = \dfrac{ 2 v}{z}H ^{( 2 )} _{v} (z,w) - \dfrac{ 2 \tanh vw}{z}\dfrac{\partial}{\partial w}H ^{( 2 )} _{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:^[1] K _{v(z,w)=\int} ^{inf ty} ^{w} e ^{-z\cosh} \cosh vt~dt J _{v(z,w)=\int} ^{we} ^{0} ^{-z\cosh} \cosh vt~dt

Properties

J _{v(z,w)=J} ^{(v\pi i)/(2)} ^{v(z) + \dfrac{e} J(iz,v,w) - e ^{- (v\pi i)/(2)} J( - iz,v,w)}{i\pi} Y _{v(z,w)=Y} ^{(v\pi i)/(2)} ^{v(z) + \dfrac{e} J(iz,v,w) + e ^{- (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\pi i} ^{v(z) + \dfrac{J( - z,v,w) - e} J(z,v,w)}{i\pi} I ^{- (v\pi i)/(2)} _{v(z,w) = e} 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 ^{( 1 )} _{v} (z,w) = J _{v(z,w)+iY} _{v(z,w)} H ^{( 2 )} _{v} (z,w) = J _{v(z,w)-iY} _{v(z,w)} H ^{( 1 )} _{- v} (z,w) = e ^{v\pi} H ^{( 1 )} _{v} (z,w) H ^{( 2 )} _{- v} (z,w) = e ^{-v\pi} H ^{( 2 )} _{v} (z,w) H ^{( 1 )} _{v} (z,w) = \dfrac{J _{-v} (z,w) - e ^{-v\pi} J _{v(z,w)}{i\sin} v\pi} = \dfrac{Y _{-v} (z,w) - e ^{-v\pi} Y _{v(z,w)}{\sin} v\pi} for non-integer v H ^{( 2 )} _{v} (z,w) = \dfrac{e ^{v\pi} J _{v(z,w)-J} _{-v} (z,w)}{i\sin v\pi} = \dfrac{Y _{-v} (z,w) - e ^{v\pi} Y _{v(z,w)}{\sin} v\pi} for non-integer v

Differential equations

K _{v(z,w)} satisfies the inhomogeneous Bessel's differential equationz ^{2\dfrac{d} ^{2y}{dz} ^{2}+z\dfrac{dy}{dz}-(x} ^{2+v} ^{2)y=(v\sinh} vw + z\cosh vw\sinh w)e ^{-z\cosh} Both J _{v(z,w)} , Y _{v(z,w)} , H ^{( 1 )} _{v} (z,w) and H ^{( 2 )} _{v} (z,w) satisfy the partial differential equationz ^{2\dfrac{\partial} ^{2y}{\partial} z ^{2}+z\dfrac{\partial} y}{\partial z} + (z ^{2-v} ^{2)y-\dfrac{\partial} ^{2y}{\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 equationz ^{2\dfrac{\partial} ^{2y}{\partial} z ^{2}+z\dfrac{\partial} y}{\partial z} - (z ^{2+v} ^{2)y-\dfrac{\partial} ^{2y}{\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{align} J _{v(z,w)&=J} _{v(z)+\dfrac{1}{\pi} i}\left(\int ^{we} _{0} ^{(v\pi i)/(- iz\cosh t 2)} \cosh vt~dt - \int ^{we} _{0} ^{iz\cosh t - (v\pi i)/(2)} \cosh vt~dt\right) \\& = J _{v(z)+\dfrac{1}{\pi} i}\left(\int ^{w\cos\left(z\cosh} _{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt - i\int ^{w\sin\left(z\cosh} _{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt\right.\\ & \left. - \int ^{w\cos\left(z\cosh} _{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt - i\int ^{w\sin\left(z\cosh} _{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt\right) \\& = J _{v(z)+\dfrac{1}{\pi} i}\left( -2 i\int ^{w\sin\left(z\cosh} _{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt\right) \\& = J _{v(z)-\dfrac{2}{\pi}\int} ^{w\sin\left(z\cosh} ^{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt\end{align} \begin{align} Y _{v(z,w)&=Y} _{v(z)+\dfrac{1}{\pi}\left(\int} ^{we} ^{0} ^{(v\pi i)/(- iz\cosh t 2)} \cosh vt~dt + \int ^{we} _{0} ^{iz\cosh t - (v\pi i)/(2)} \cosh vt~dt\right) \\& = Y _{v(z)+\dfrac{1}{\pi}\left(\int} ^{w\cos\left(z\cosh} ^{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt - i\int ^{w\sin\left(z\cosh} _{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt\right.\\ & \left. + \int ^{w\cos\left(z\cosh} _{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt + i\int ^{w\sin\left(z\cosh} _{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt\right) \\& = Y _{v(z)+\dfrac{2}{\pi}\int} ^{w\cos\left(z\cosh} ^{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt\end{align} With the Mehler–Sonine integral expressions of J _{v(z)=\dfrac{2}{\pi}\int} ^{inf ty\sin\left(z\cosh} ^{0} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt and Y _{v(z)=-\dfrac{2}{\pi}\int} ^{inf ty\cos\left(z\cosh} ^{0} 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} ^{inf ty\sin\left(z\cosh} ^{w} t - \dfrac{v\pi}{ 2 }\right)\cosh vt~dt and Y _{v(z,w)=-\dfrac{2}{\pi}\int} ^{inf ty\cos\left(z\cosh} ^{w} 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 .

External links

• Book: Agrest . Matest M. . Maksimov . Michail S. . Theory of Incomplete Cylindrical Functions and their Applications . 1971 . Springer-Verlag Berlin Heidelberg . Berlin, Heidelberg . 978-3-642-65023-9.
• Cicchetti . R. . Faraone . A. . Incomplete Hankel and Modified Bessel Functions: A Class of Special Functions for Electromagnetics . IEEE Transactions on Antennas and Propagation . December 2004 . 52 . 12 . 3373–3389 . 10.1109/TAP.2004.835269. 2004ITAP...52.3373C . 25089438 .
• Jones . D. S. . Incomplete Bessel functions. II. Asymptotic expansions for large argument . Proceedings of the Edinburgh Mathematical Society . October 2007 . 50 . 3 . 711–723 . 10.1017/S0013091505000908. free .

Notes and References

1. Jones . D. S. . Incomplete Bessel functions. I . Proceedings of the Edinburgh Mathematical Society . February 2007 . 50 . 1 . 173–183 . 10.1017/S0013091505000490 . free .