Incomplete Bessel functions
(Learn how and when to remove this message) 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:
![{\displaystyle J_{v-1}(z,w)-J_{v+1}(z,w)=2{\dfrac {\partial }{\partial z}}J_{v}(z,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69c90f422c17b5c086a658442d5476843676155c)
![{\displaystyle Y_{v-1}(z,w)-Y_{v+1}(z,w)=2{\dfrac {\partial }{\partial z}}Y_{v}(z,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/248be1d906db878a3be21b89bb373f0c538ccca1)
![{\displaystyle I_{v-1}(z,w)+I_{v+1}(z,w)=2{\dfrac {\partial }{\partial z}}I_{v}(z,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60e6a8e67d8323952fb555676fe4634466ae4f63)
![{\displaystyle K_{v-1}(z,w)+K_{v+1}(z,w)=-2{\dfrac {\partial }{\partial z}}K_{v}(z,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a02293784b6854ffbd8b33daab1373053c56b5e6)
![{\displaystyle H_{v-1}^{(1)}(z,w)-H_{v+1}^{(1)}(z,w)=2{\dfrac {\partial }{\partial z}}H_{v}^{(1)}(z,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0b8ec189c1b1bb46274411ad94ff225147990d)
![{\displaystyle H_{v-1}^{(2)}(z,w)-H_{v+1}^{(2)}(z,w)=2{\dfrac {\partial }{\partial z}}H_{v}^{(2)}(z,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c74a40aaac760df0dcf25b7567a3a8874e60237)
And the following suitable extension forms of delay differential equations from that of the complete-type Bessel functions:
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3097abdb920667c10689e8d6b7d0ee2b91a3f0fc)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28b982a60ec3c55f099399bffe21430c5a88ad66)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e60a3932c0c36367aefbe4ebaf61bc014e85b4ca)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14766cefaefc78e77be04b314463202578473d32)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64e31e2b864ebcffaacb6aecaf61832a023694bb)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa00b43ba367d2635b280233535a42efd305678)
Where the new parameter
defines the integral bound of the upper-incomplete form and lower-incomplete form of the modified Bessel function of the second kind:[1]
![{\displaystyle K_{v}(z,w)=\int _{w}^{\infty }e^{-z\cosh t}\cosh vt~dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/385655ee9a54a76cad5252ff40f1288eb54aa0af)
![{\displaystyle J_{v}(z,w)=\int _{0}^{w}e^{-z\cosh t}\cosh vt~dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34d29e1e5c5e77610158b47c9be753868b0a6a07)
Properties
![{\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 }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/700ff885023257d3e8487adda7c49f3814fd3fd8)
![{\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 }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d22ca3378e090912054c21f77e356166cf6a4c3f)
for integer ![{\displaystyle v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf81c8b801a86685a725abfe3390589fb325858)
![{\displaystyle I_{v}(z,w)=I_{v}(z)+{\dfrac {J(-z,v,w)-e^{-v\pi i}J(z,v,w)}{i\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf630f787cc7964ca6921ad4530a6cee23a2eb6)
![{\displaystyle I_{v}(z,w)=e^{-{\frac {v\pi i}{2}}}J_{v}(iz,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/360b88b0a4490d3d0c3c972374612f33c43a0388)
![{\displaystyle K_{-v}(z,w)=K_{v}(z,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ee229a585f5034336c6891e58b320308b4c577)
for non-integer ![{\displaystyle v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
![{\displaystyle H_{v}^{(1)}(z,w)=J_{v}(z,w)+iY_{v}(z,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29105fcb43529aa152bb100d13a4b0f08eb4064c)
![{\displaystyle H_{v}^{(2)}(z,w)=J_{v}(z,w)-iY_{v}(z,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/602ae5d6636d2ab9cc92e9bab056ee377fa7588c)
![{\displaystyle H_{-v}^{(1)}(z,w)=e^{v\pi i}H_{v}^{(1)}(z,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2228572ab5695c7ae70199e14ea519e5bb53d0bc)
![{\displaystyle H_{-v}^{(2)}(z,w)=e^{-v\pi i}H_{v}^{(2)}(z,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cae1499c60f3d792a21da2b8be9a4f2520a20656)
for non-integer ![{\displaystyle v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
for non-integer ![{\displaystyle v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
Differential equations
satisfies the inhomogeneous Bessel's differential equation
![{\displaystyle z^{2}{\dfrac {d^{2}y}{dz^{2}}}+z{\dfrac {dy}{dz}}-(x^{2}+v^{2})y=(v\sinh vw+z\cosh vw\sinh w)e^{-z\cosh w}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d13c22b8786cf4496f7264a387a312c92f82eab)
Both
,
,
and
satisfy the partial differential equation
![{\displaystyle z^{2}{\dfrac {\partial ^{2}y}{\partial z^{2}}}+z{\dfrac {\partial y}{\partial z}}+(z^{2}-v^{2})y-{\dfrac {\partial ^{2}y}{\partial w^{2}}}+2v\tanh vw{\dfrac {\partial y}{\partial w}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d81728ddd08e621383804fa23dd42e59754c05)
Both
and
satisfy the partial differential equation
![{\displaystyle z^{2}{\dfrac {\partial ^{2}y}{\partial z^{2}}}+z{\dfrac {\partial y}{\partial z}}-(z^{2}+v^{2})y-{\dfrac {\partial ^{2}y}{\partial w^{2}}}+2v\tanh vw{\dfrac {\partial y}{\partial w}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b42f621bc25942270783fb82752ba8e9bd9da88)
Integral representations
Base on the preliminary definitions above, one would derive directly the following integral forms of
,
:
![{\displaystyle {\begin{aligned}J_{v}(z,w)&=J_{v}(z)+{\dfrac {1}{\pi i}}\left(\int _{0}^{w}e^{{\frac {v\pi i}{2}}-iz\cosh t}\cosh vt~dt-\int _{0}^{w}e^{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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22967215533f0f9e04596b8ab476a7a8d0202a90)
![{\displaystyle {\begin{aligned}Y_{v}(z,w)&=Y_{v}(z)+{\dfrac {1}{\pi }}\left(\int _{0}^{w}e^{{\frac {v\pi i}{2}}-iz\cosh t}\cosh vt~dt+\int _{0}^{w}e^{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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bca5c6ce359d3679aa64d98f81399de41d78225c)
With the Mehler–Sonine integral expressions of
and
mentioned in Digital Library of Mathematical Functions,[2]
we can further simplify to
and
, but the issue is not quite good since the convergence range will reduce greatly to
.
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.