Основне функције
уреди
Рекурзивне релације
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Рекурзивне релације за модификоване функције првога реда
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m
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I
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z
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I
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{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{\nu }I_{\nu }(z){\Bigr ]}=z^{\nu -m}I_{\nu -m}(z).}
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{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{-\nu }I_{\nu }(z){\Bigr ]}=z^{-\nu -m}I_{\nu +m}(z).}
I
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I
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{\displaystyle ~I_{\nu -1}(z)-I_{\nu +1}(z)=2\nu z^{-1}I_{\nu }(z).}
I
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I
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′
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{\displaystyle ~I_{\nu -1}(z)+I_{\nu +1}(z)=2I'_{\nu }(z).}
Рекурзивне релације за модификоване функције другога реда
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d
z
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m
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K
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K
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{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{\nu }K_{\nu }(z){\Bigr ]}=(-1)^{m}z^{\nu -m}K_{\nu -m}(z).}
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{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{-\nu }K_{\nu }(z){\Bigr ]}=(-1)^{m}z^{-\nu -m}K_{\nu +m}(z).}
K
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1
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1
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−
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ν
z
−
1
K
ν
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z
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{\displaystyle ~K_{\nu -1}(z)-K_{\nu +1}(z)=-2\nu z^{-1}K_{\nu }(z).}
K
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1
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K
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+
1
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=
−
2
K
ν
′
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z
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{\displaystyle ~K_{\nu -1}(z)+K_{\nu +1}(z)=-2K'_{\nu }(z).}
Вронскијан
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W
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I
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I
−
ν
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z
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=
−
2
sin
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ν
π
)
π
z
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{\displaystyle W\left[I_{\nu }(z),I_{-\nu }(z)\right]=-{\frac {2\sin(\nu \pi )}{\pi z}}.}
W
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K
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−
z
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1
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{\displaystyle W\left[I_{\nu }(z),K_{\nu }(z)\right]=-z^{-1}.}
Интегрални приказ
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Интегрални приказ модификованих функција првога реда
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I
ν
(
z
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=
2
−
ν
z
ν
π
Γ
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ν
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2
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∫
0
π
e
z
cos
t
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sin
t
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2
ν
d
t
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R
e
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ν
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>
−
1
2
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Γ
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z
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{\displaystyle I_{\nu }(z)={\frac {2^{-\nu }z^{\nu }}{{\sqrt {\pi }}\Gamma (\nu +{\frac {1}{2}})}}\int _{0}^{\pi }e^{z\cos t}\left(\sin t\right)^{2\nu }dt,\qquad Re(\nu )>-{\frac {1}{2}},\Gamma (z)}
гама функција .
I
ν
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z
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=
2
1
−
ν
z
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π
Γ
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ν
+
1
2
)
∫
0
1
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1
−
t
2
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ν
−
1
2
cosh
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z
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d
t
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R
e
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2
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{\displaystyle I_{\nu }(z)={\frac {2^{1-\nu }z^{\nu }}{{\sqrt {\pi }}\Gamma (\nu +{\frac {1}{2}})}}\int _{0}^{1}(1-t^{2})^{\nu -{\frac {1}{2}}}\cosh(zt)dt,\qquad Re(\nu )>-{\frac {1}{2}}.}
I
ν
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z
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=
2
−
ν
z
ν
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Γ
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2
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∫
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1
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{\displaystyle I_{\nu }(z)={\frac {2^{-\nu }z^{\nu }}{{\sqrt {\pi }}\Gamma (\nu +{\frac {1}{2}})}}\int _{-1}^{1}(1-t^{2})^{\nu -{\frac {1}{2}}}e^{-zt}dt,\qquad Re(\nu )>-{\frac {1}{2}}.}
I
n
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z
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=
1
π
∫
0
π
e
z
cos
t
cos
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n
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)
d
t
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n
∈
Z
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R
e
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>
0.
{\displaystyle I_{n}(z)={\frac {1}{\pi }}\int _{0}^{\pi }e^{z\cos t}\cos(nt)dt,\qquad n\in \mathbb {Z} ,Re(z)>0.}
Интегрални приказ модификованих функција другога реда
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K
ν
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z
)
=
∫
0
∞
e
−
z
cosh
t
cosh
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ν
t
)
d
t
,
|
A
r
g
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z
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|
<
π
2
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{\displaystyle {{K}_{\nu }}(z)=\int _{0}^{\infty }{{e}^{-z\cosh t}}\cosh(\nu t)dt,\qquad |Arg(z)|<{\frac {\pi }{2}}.}
K
ν
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π
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Γ
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∫
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∞
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ν
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d
t
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>
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1
2
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|
A
r
g
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z
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<
π
2
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{\displaystyle {{K}_{\nu }}(z)={\frac {{\sqrt {\pi }}{{\left({\frac {z}{2}}\right)}^{\nu }}}{\Gamma (\nu +{\frac {1}{2}})}}\int _{1}^{\infty }{{({{t}^{2}}-1)}^{\nu -{\frac {1}{2}}}}{{e}^{-zt}}dt,\qquad Re(\nu )>-{\frac {1}{2}},|Arg(z)|<{\frac {\pi }{2}}.}
K
ν
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=
π
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z
2
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ν
Γ
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1
2
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∫
0
∞
e
−
z
cosh
t
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sinh
t
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2
ν
d
t
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R
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>
−
1
2
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|
A
r
g
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<
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2
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{\displaystyle {{K}_{\nu }}(z)={\frac {{\sqrt {\pi }}{{\left({\frac {z}{2}}\right)}^{\nu }}}{\Gamma (\nu +{\frac {1}{2}})}}\int _{0}^{\infty }{{e}^{-z\cosh t}}{{\left(\sinh t\right)}^{2\nu }}dt,\qquad Re(\nu )>-{\frac {1}{2}},|Arg(z)|<{\frac {\pi }{2}}.}
Асимптотски облик
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I
ν
(
z
)
∝
e
z
2
π
z
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1
+
O
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1
z
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)
,
|
A
r
g
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z
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<
π
2
,
|
z
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→
∞
.
{\displaystyle I_{\nu }(z)\varpropto {\frac {e^{z}}{\sqrt {2\pi z}}}\left(1+O\left({\frac {1}{z}}\right)\right),\qquad \left|Arg(z)\right|<{\frac {\pi }{2}},\left|z\right|\to \infty .}
K
ν
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z
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∝
π
2
e
−
z
z
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1
+
O
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1
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,
|
z
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→
∞
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{\displaystyle K_{\nu }(z)\varpropto {\sqrt {\frac {\pi }{2}}}{\frac {e^{-z}}{\sqrt {z}}}\left(1+O\left({\frac {1}{z}}\right)\right),\qquad \left|z\right|\to \infty .}
Литература
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![{\displaystyle ~K_{\nu }(x)={\frac {\pi }{2\sin \nu \pi }}{\biggl [}I_{\nu }(x)-I_{-\nu }(x){\biggr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41afac4736bafb7b164566006ca9b1fa24e13b59)
![{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{\nu }I_{\nu }(z){\Bigr ]}=z^{\nu -m}I_{\nu -m}(z).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9bd25915d6648205051d5508bcc2b680b2f0b72)
![{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{-\nu }I_{\nu }(z){\Bigr ]}=z^{-\nu -m}I_{\nu +m}(z).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d6d5572ab3f3a29b1e88e0a66ef4fcb0b63bf8)
![{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{\nu }K_{\nu }(z){\Bigr ]}=(-1)^{m}z^{\nu -m}K_{\nu -m}(z).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31958576622e5d2744cf287d0f36172b9542ce29)
![{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{-\nu }K_{\nu }(z){\Bigr ]}=(-1)^{m}z^{-\nu -m}K_{\nu +m}(z).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc5049e631bdb98112fcfc49c9b0b86c724db6b)
![{\displaystyle W\left[I_{\nu }(z),I_{-\nu }(z)\right]=-{\frac {2\sin(\nu \pi )}{\pi z}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f57830f12153346dcf1d345485e72c72c8cc5d)
![{\displaystyle W\left[I_{\nu }(z),K_{\nu }(z)\right]=-z^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6562d5a94c6b89151aeeb16f23a66da710fa596d)
