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:

[math]\displaystyle{ J_{v-1}(z,w)-J_{v+1}(z,w)=2\dfrac{\partial}{\partial z}J_v(z,w) }[/math]

[math]\displaystyle{ Y_{v-1}(z,w)-Y_{v+1}(z,w)=2\dfrac{\partial}{\partial z}Y_v(z,w) }[/math]

[math]\displaystyle{ I_{v-1}(z,w)+I_{v+1}(z,w)=2\dfrac{\partial}{\partial z}I_v(z,w) }[/math]

[math]\displaystyle{ K_{v-1}(z,w)+K_{v+1}(z,w)=-2\dfrac{\partial}{\partial z}K_v(z,w) }[/math]

[math]\displaystyle{ H_{v-1}^{(1)}(z,w)-H_{v+1}^{(1)}(z,w)=2\dfrac{\partial}{\partial z}H_v^{(1)}(z,w) }[/math]

[math]\displaystyle{ H_{v-1}^{(2)}(z,w)-H_{v+1}^{(2)}(z,w)=2\dfrac{\partial}{\partial z}H_v^{(2)}(z,w) }[/math]

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

[math]\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) }[/math]

[math]\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) }[/math]

[math]\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) }[/math]

[math]\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) }[/math]

[math]\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) }[/math]

[math]\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) }[/math]

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

[math]\displaystyle{ K_v(z,w)=\int_w^\infty e^{-z\cosh t}\cosh vt~dt }[/math]

[math]\displaystyle{ J_v(z,w)=\int_0^we^{-z\cosh t}\cosh vt~dt }[/math]

Properties

[math]\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} }[/math]

[math]\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} }[/math]

[math]\displaystyle{ I_{-v}(z,w)=I_v(z,w) }[/math] for integer [math]\displaystyle{ v }[/math]

[math]\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) }[/math]

[math]\displaystyle{ I_v(z,w)=I_v(z)+\dfrac{J(-z,v,w)-e^{-v\pi i}J(z,v,w)}{i\pi} }[/math]

[math]\displaystyle{ I_v(z,w)=e^{-\frac{v\pi i}{2}}J_v(iz,w) }[/math]

[math]\displaystyle{ K_{-v}(z,w)=K_v(z,w) }[/math]

[math]\displaystyle{ K_v(z,w)=\dfrac{\pi}{2}\dfrac{I_{-v}(z,w)-I_v(z,w)}{\sin v\pi} }[/math] for non-integer [math]\displaystyle{ v }[/math]

[math]\displaystyle{ H_v^{(1)}(z,w)=J_v(z,w)+iY_v(z,w) }[/math]

[math]\displaystyle{ H_v^{(2)}(z,w)=J_v(z,w)-iY_v(z,w) }[/math]

[math]\displaystyle{ H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w) }[/math]

[math]\displaystyle{ H_{-v}^{(2)}(z,w)=e^{-v\pi i}H_v^{(2)}(z,w) }[/math]

[math]\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} }[/math] for non-integer [math]\displaystyle{ v }[/math]

[math]\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} }[/math] for non-integer [math]\displaystyle{ v }[/math]

Differential equations

[math]\displaystyle{ K_v(z,w) }[/math] satisfies the inhomogeneous Bessel's differential equation

[math]\displaystyle{ z^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 w} }[/math]

Both [math]\displaystyle{ J_v(z,w) }[/math] , [math]\displaystyle{ Y_v(z,w) }[/math] , [math]\displaystyle{ H_v^{(1)}(z,w) }[/math] and [math]\displaystyle{ H_v^{(2)}(z,w) }[/math] satisfy the partial differential equation

[math]\displaystyle{ z^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 }[/math]

Both [math]\displaystyle{ I_v(z,w) }[/math] and [math]\displaystyle{ K_v(z,w) }[/math] satisfy the partial differential equation

[math]\displaystyle{ z^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 }[/math]

Integral representations

Base on the preliminary definitions above, one would derive directly the following integral forms of [math]\displaystyle{ J_v(z,w) }[/math] , [math]\displaystyle{ Y_v(z,w) }[/math]:

[math]\displaystyle{ \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-\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} }[/math]

[math]\displaystyle{ \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+\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} }[/math]

With the Mehler–Sonine integral expressions of [math]\displaystyle{ J_v(z)=\dfrac{2}{\pi}\int_0^\infty\sin\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt }[/math] and [math]\displaystyle{ Y_v(z)=-\dfrac{2}{\pi}\int_0^\infty\cos\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt }[/math] mentioned in Digital Library of Mathematical Functions,

we can further simplify to [math]\displaystyle{ J_v(z,w)=\dfrac{2}{\pi}\int_w^\infty\sin\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt }[/math] and [math]\displaystyle{ Y_v(z,w)=-\dfrac{2}{\pi}\int_w^\infty\cos\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt }[/math] , but the issue is not quite good since the convergence range will reduce greatly to [math]\displaystyle{ |v|\lt 1 }[/math].

External links

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