Some mathematicians defined this type incomplete-version of Bessel function or this type generalized-version of incomplete gamma function :[1] [2] [3] [4] [5]
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
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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 [6] 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
↑ "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 .
↑ "Archived copy" (PDF) . S2CID 126117454 . Archived from the original (PDF) on 2019-12-23. Retrieved 2019-12-23 . CS1 maint: archived copy as title (link )
↑ 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