From Wikipedia, the free encyclopedia
Some mathematicians defined this type incomplete-version of
Bessel function or this type generalized-version of
incomplete gamma function:
[1]
[2]
[3]
[4]
[5]
![{\displaystyle K_{v}(x,y)=\int _{1}^{\infty }{\frac {e^{-xt-{\frac {y}{t}}}}{t^{v+1}}}~dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8920a30919e4a1117276172bf3a9036074b423db)
![{\displaystyle \gamma (\alpha ,x;b)=\int _{0}^{x}t^{\alpha -1}e^{-t-{\frac {b}{t}}}~dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f545d6c534e0995a289d165a5efe704a80d87bad)
![{\displaystyle \Gamma (\alpha ,x;b)=\int _{x}^{\infty }t^{\alpha -1}e^{-t-{\frac {b}{t}}}~dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3de1d5fd51be7f1d2a2ad3e02fcb4699d531dd7e)
Properties
![{\displaystyle K_{v}(x,y)=x^{v}\Gamma (-v,x;xy)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b9b58d08e493b7fb77afd2c1240bdd4b2448d0)
![{\displaystyle K_{v}(x,y)+K_{-v}(y,x)={\frac {2x^{\frac {v}{2}}}{y^{\frac {v}{2}}}}K_{v}(2{\sqrt {xy}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8d45b1f5fb4afbb57f01ad0eb698009be29af)
![{\displaystyle \gamma (\alpha ,x;0)=\gamma (\alpha ,x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f68ff5bd3cd89669132ec5034905238d5f60c6c)
![{\displaystyle \Gamma (\alpha ,x;0)=\Gamma (\alpha ,x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4aff0ee8ec171aa7508b9b91b39f82089ec01823)
![{\displaystyle \gamma (\alpha ,x;b)+\Gamma (\alpha ,x;b)=2b^{\frac {\alpha }{2}}K_{\alpha }(2{\sqrt {b}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/349ec08f2d9ee4c215a813688c7ed62d1e678ab6)
One of the advantage of defining this type incomplete-version of
Bessel function
is that even for example the associated Anger–Weber function defined in
Digital Library of Mathematical Functions
[6] can related:
![{\displaystyle \mathbf {A} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\infty }e^{-\nu t-z\sinh t}~dt={\frac {1}{\pi }}\int _{0}^{\infty }e^{-(\nu +1)t-{\frac {ze^{t}}{2}}+{\frac {z}{2e^{t}}}}~d(e^{t})={\frac {1}{\pi }}\int _{1}^{\infty }{\frac {e^{-{\frac {zt}{2}}+{\frac {z}{2t}}}}{t^{\nu +1}}}~dt={\frac {1}{\pi }}K_{\nu }\left({\frac {z}{2}},-{\frac {z}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a71263a1d40600941510f2c58603e1e6b1d3bd1)
Recurrence relations
satisfy this
recurrence relation:
![{\displaystyle xK_{v-1}(x,y)+vK_{v}(x,y)-yK_{v+1}(x,y)=e^{-x-y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6171d44fe6e8507022f484403250512278bf900)
References
-
^
"incompleteBesselK function | R Documentation". www.rdocumentation.org.
-
^
"incompleteBesselK: The Incomplete Bessel K Function in DistributionUtils: Distribution Utilities". rdrr.io.
-
^ Harris, Frank E. (2008).
"Incomplete Bessel, generalized incomplete gamma, or leaky aquifer functions" (PDF). Journal of Computational and Applied Mathematics. 215: 260–269.
doi:
10.1016/j.cam.2007.04.008. Retrieved 2020-01-08.
-
^
"Generalized incomplete gamma function and its application". 2018-01-14. Retrieved 2020-01-08.
-
^ Didem Aşçıoğlu (September 2015).
The Generalized Incomplete Gamma Functions (PDF) (Master thesis). Eastern Mediterranean University.
S2CID
126117454. Archived from
the original (PDF) on 2019-12-23. Retrieved 2019-12-23 – via Semantic Scholar.
-
^ Paris, R. B. (2010),
"Anger-Weber 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.