Misplaced Pages

Incomplete Bessel K function/generalized incomplete gamma function

Article snapshot taken from[REDACTED] with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

Some mathematicians defined this type incomplete-version of Bessel function or this type generalized-version of incomplete gamma function:

K v ( x , y ) = ∫ 1 ∞ e − x t − y t t v + 1   d t {\displaystyle K_{v}(x,y)=\int _{1}^{\infty }{\frac {e^{-xt-{\frac {y}{t}}}}{t^{v+1}}}~dt}
γ ( α , x ; b ) = ∫ 0 x t α − 1 e − t − b t   d t {\displaystyle \gamma (\alpha ,x;b)=\int _{0}^{x}t^{\alpha -1}e^{-t-{\frac {b}{t}}}~dt}
Γ ( α , x ; b ) = ∫ x ∞ t α − 1 e − t − b t   d t {\displaystyle \Gamma (\alpha ,x;b)=\int _{x}^{\infty }t^{\alpha -1}e^{-t-{\frac {b}{t}}}~dt}

Properties

K v ( x , y ) = x v Γ ( − v , x ; x y ) {\displaystyle K_{v}(x,y)=x^{v}\Gamma (-v,x;xy)}
K v ( x , y ) + K − v ( y , x ) = 2 x v 2 y v 2 K v ( 2 x y ) {\displaystyle K_{v}(x,y)+K_{-v}(y,x)={\frac {2x^{\frac {v}{2}}}{y^{\frac {v}{2}}}}K_{v}(2{\sqrt {xy}})}
γ ( α , x ; 0 ) = γ ( α , x ) {\displaystyle \gamma (\alpha ,x;0)=\gamma (\alpha ,x)}
Γ ( α , x ; 0 ) = Γ ( α , x ) {\displaystyle \Gamma (\alpha ,x;0)=\Gamma (\alpha ,x)}
γ ( α , x ; b ) + Γ ( α , x ; b ) = 2 b α 2 K α ( 2 b ) {\displaystyle \gamma (\alpha ,x;b)+\Gamma (\alpha ,x;b)=2b^{\frac {\alpha }{2}}K_{\alpha }(2{\sqrt {b}})}

One of the advantage of defining this type incomplete-version of Bessel function K v ( x , y ) {\displaystyle K_{v}(x,y)} is that even for example the associated Anger–Weber function defined in Digital Library of Mathematical Functions can related:

A ν ( z ) = 1 π ∫ 0 ∞ e − ν t − z sinh ⁡ t   d t = 1 π ∫ 0 ∞ e − ( ν + 1 ) t − z e t 2 + z 2 e t   d ( e t ) = 1 π ∫ 1 ∞ e − z t 2 + z 2 t t ν + 1   d t = 1 π K ν ( z 2 , − z 2 ) {\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)}

Recurrence relations

K v ( x , y ) {\displaystyle K_{v}(x,y)} satisfy this recurrence relation:

x K v − 1 ( x , y ) + v K v ( x , y ) − y K v + 1 ( x , y ) = e − x − y {\displaystyle xK_{v-1}(x,y)+vK_{v}(x,y)-yK_{v+1}(x,y)=e^{-x-y}}

References

  1. "incompleteBesselK function | R Documentation". www.rdocumentation.org.
  2. "incompleteBesselK: The Incomplete Bessel K Function in DistributionUtils: Distribution Utilities". rdrr.io.
  3. 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.
  4. "Generalized incomplete gamma function and its application". 2018-01-14. Retrieved 2020-01-08.
  5. 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.
  6. 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.
Categories:
Incomplete Bessel K function/generalized incomplete gamma function Add topic