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{2 v}{z} J_{v} (z, w) - \frac{2 \tanh v w}{z} \frac{\partial}{\partial w} J_{v} (z, w)

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

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

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

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

H_{v - 1}^{(2)} (z, w) + H_{v + 1}^{(2)} (z, w) = \frac{2 v}{z} H_{v}^{(2)} (z, w) - \frac{2 \tanh v w}{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) = \int_{w}^{\infty} e^{- z \cosh t} \cosh v t d t

J_{v} (z, w) = \int_{0}^{w} e^{- z \cosh t} \cosh v t d t

Properties

J_{v} (z, w) = J_{v} (z) + \frac{e^{\frac{v π i}{2}} J (i z, v, w) - e^{- \frac{v π i}{2}} J (- i z, v, w)}{i π}

Y_{v} (z, w) = Y_{v} (z) + \frac{e^{\frac{v π i}{2}} J (i z, v, w) + e^{- \frac{v π i}{2}} J (- i z, v, w)}{π}

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 π}{π} J (z, v, w)

I_{v} (z, w) = I_{v} (z) + \frac{J (- z, v, w) - e^{- v π i} J (z, v, w)}{i π}

I_{v} (z, w) = e^{- \frac{v π i}{2}} J_{v} (i z, w)

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

K_{v} (z, w) = \frac{π}{2} \frac{I_{- v} (z, w) - I_{v} (z, w)}{\sin v π}

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 π i} H_{v}^{(1)} (z, w)

H_{- v}^{(2)} (z, w) = e^{- v π i} H_{v}^{(2)} (z, w)

H_{v}^{(1)} (z, w) = \frac{J_{- v} (z, w) - e^{- v π i} J_{v} (z, w)}{i \sin v π} = \frac{Y_{- v} (z, w) - e^{- v π i} Y_{v} (z, w)}{\sin v π}

for non-integer

v

H_{v}^{(2)} (z, w) = \frac{e^{v π i} J_{v} (z, w) - J_{- v} (z, w)}{i \sin v π} = \frac{Y_{- v} (z, w) - e^{v π i} Y_{v} (z, w)}{\sin v π}

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{d y}{d z} - (x^{2} + v^{2}) y = (v \sinh v w + z \cosh v w \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}} + 2 v \tanh v w \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}} + 2 v \tanh v w \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{matrix} J_{v} (z, w) & = J_{v} (z) + \frac{1}{π i} (\int_{0}^{w} e^{\frac{v π i}{2} - i z \cosh t} \cosh v t d t - \int_{0}^{w} e^{i z \cosh t - \frac{v π i}{2}} \cosh v t d t) \\ = J_{v} (z) + \frac{1}{π i} (\int_{0}^{w} \cos (z \cosh t - \frac{v π}{2}) \cosh v t d t - i \int_{0}^{w} \sin (z \cosh t - \frac{v π}{2}) \cosh v t d t \\ - \int_{0}^{w} \cos (z \cosh t - \frac{v π}{2}) \cosh v t d t - i \int_{0}^{w} \sin (z \cosh t - \frac{v π}{2}) \cosh v t d t) \\ = J_{v} (z) + \frac{1}{π i} (- 2 i \int_{0}^{w} \sin (z \cosh t - \frac{v π}{2}) \cosh v t d t) \\ = J_{v} (z) - \frac{2}{π} \int_{0}^{w} \sin (z \cosh t - \frac{v π}{2}) \cosh v t d t \end{matrix}

\begin{matrix} Y_{v} (z, w) & = Y_{v} (z) + \frac{1}{π} (\int_{0}^{w} e^{\frac{v π i}{2} - i z \cosh t} \cosh v t d t + \int_{0}^{w} e^{i z \cosh t - \frac{v π i}{2}} \cosh v t d t) \\ = Y_{v} (z) + \frac{1}{π} (\int_{0}^{w} \cos (z \cosh t - \frac{v π}{2}) \cosh v t d t - i \int_{0}^{w} \sin (z \cosh t - \frac{v π}{2}) \cosh v t d t \\ + \int_{0}^{w} \cos (z \cosh t - \frac{v π}{2}) \cosh v t d t + i \int_{0}^{w} \sin (z \cosh t - \frac{v π}{2}) \cosh v t d t) \\ = Y_{v} (z) + \frac{2}{π} \int_{0}^{w} \cos (z \cosh t - \frac{v π}{2}) \cosh v t d t \end{matrix}

With the Mehler–Sonine integral expressions of $J_{v} (z) = \frac{2}{π} \int_{0}^{\infty} \sin (z \cosh t - \frac{v π}{2}) \cosh v t d t$ and $Y_{v} (z) = - \frac{2}{π} \int_{0}^{\infty} \cos (z \cosh t - \frac{v π}{2}) \cosh v t d t$ mentioned in Digital Library of Mathematical Functions,

we can further simplify to $J_{v} (z, w) = \frac{2}{π} \int_{w}^{\infty} \sin (z \cosh t - \frac{v π}{2}) \cosh v t d t$ and $Y_{v} (z, w) = - \frac{2}{π} \int_{w}^{\infty} \cos (z \cosh t - \frac{v π}{2}) \cosh v t d t$ , but the issue is not quite good since the convergence range will reduce greatly to $| v | < 1$ .

References

↑ Jones, D. S. (February 2007). "Incomplete Bessel functions. I". Proceedings of the Edinburgh Mathematical Society. 50 (1): 173–183. doi:10.1017/S0013091505000490.
↑ Paris, R. B. (2010), "Bessel Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

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. Bibcode:2004ITAP...52.3373C. doi:10.1109/TAP.2004.835269. S2CID 25089438.
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.

[1] Jones, D. S. (February 2007). "Incomplete Bessel functions. I". Proceedings of the Edinburgh Mathematical Society. 50 (1): 173–183. doi:10.1017/S0013091505000490.

[2] Paris, R. B. (2010), "Bessel Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.