Osnovne funkcije
uredi
Rekurzivne relacije
uredi
Rekurzivne relacije za modifikovane funkcije prvoga reda
uredi
(
d
z
d
z
)
m
[
z
ν
I
ν
(
z
)
]
=
z
ν
−
m
I
ν
−
m
(
z
)
.
{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{\nu }I_{\nu }(z){\Bigr ]}=z^{\nu -m}I_{\nu -m}(z).}
(
d
z
d
z
)
m
[
z
−
ν
I
ν
(
z
)
]
=
z
−
ν
−
m
I
ν
+
m
(
z
)
.
{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{-\nu }I_{\nu }(z){\Bigr ]}=z^{-\nu -m}I_{\nu +m}(z).}
I
ν
−
1
(
z
)
−
I
ν
+
1
(
z
)
=
2
ν
z
−
1
I
ν
(
z
)
.
{\displaystyle ~I_{\nu -1}(z)-I_{\nu +1}(z)=2\nu z^{-1}I_{\nu }(z).}
I
ν
−
1
(
z
)
+
I
ν
+
1
(
z
)
=
2
I
ν
′
(
z
)
.
{\displaystyle ~I_{\nu -1}(z)+I_{\nu +1}(z)=2I'_{\nu }(z).}
Rekurzivne relacije za modifikovane funkcije drugoga reda
uredi
(
d
z
d
z
)
m
[
z
ν
K
ν
(
z
)
]
=
(
−
1
)
m
z
ν
−
m
K
ν
−
m
(
z
)
.
{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{\nu }K_{\nu }(z){\Bigr ]}=(-1)^{m}z^{\nu -m}K_{\nu -m}(z).}
(
d
z
d
z
)
m
[
z
−
ν
K
ν
(
z
)
]
=
(
−
1
)
m
z
−
ν
−
m
K
ν
+
m
(
z
)
.
{\displaystyle ~\left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{-\nu }K_{\nu }(z){\Bigr ]}=(-1)^{m}z^{-\nu -m}K_{\nu +m}(z).}
K
ν
−
1
(
z
)
−
K
ν
+
1
(
z
)
=
−
2
ν
z
−
1
K
ν
(
z
)
.
{\displaystyle ~K_{\nu -1}(z)-K_{\nu +1}(z)=-2\nu z^{-1}K_{\nu }(z).}
K
ν
−
1
(
z
)
+
K
ν
+
1
(
z
)
=
−
2
K
ν
′
(
z
)
.
{\displaystyle ~K_{\nu -1}(z)+K_{\nu +1}(z)=-2K'_{\nu }(z).}
Vronskijan
uredi
W
[
I
ν
(
z
)
,
I
−
ν
(
z
)
]
=
−
2
sin
(
ν
π
)
π
z
.
{\displaystyle W\left[I_{\nu }(z),I_{-\nu }(z)\right]=-{\frac {2\sin(\nu \pi )}{\pi z}}.}
W
[
I
ν
(
z
)
,
K
ν
(
z
)
]
=
−
z
−
1
.
{\displaystyle W\left[I_{\nu }(z),K_{\nu }(z)\right]=-z^{-1}.}
Integralni prikaz
uredi
Integralni prikaz modifikovanih funkcija prvoga reda
uredi
I
ν
(
z
)
=
2
−
ν
z
ν
π
Γ
(
ν
+
1
2
)
∫
0
π
e
z
cos
t
(
sin
t
)
2
ν
d
t
,
R
e
(
ν
)
>
−
1
2
,
Γ
(
z
)
{\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)}
gama funkcija .
I
ν
(
z
)
=
2
1
−
ν
z
ν
π
Γ
(
ν
+
1
2
)
∫
0
1
(
1
−
t
2
)
ν
−
1
2
cosh
(
z
t
)
d
t
,
R
e
(
ν
)
>
−
1
2
.
{\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
(
z
)
=
2
−
ν
z
ν
π
Γ
(
ν
+
1
2
)
∫
−
1
1
(
1
−
t
2
)
ν
−
1
2
e
−
z
t
d
t
,
R
e
(
ν
)
>
−
1
2
.
{\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
(
z
)
=
1
π
∫
0
π
e
z
cos
t
cos
(
n
t
)
d
t
,
n
∈
Z
,
R
e
(
z
)
>
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.}
Integralni prikaz modifikovanih funkcija drugoga reda
uredi
K
ν
(
z
)
=
∫
0
∞
e
−
z
cosh
t
cosh
(
ν
t
)
d
t
,
|
A
r
g
(
z
)
|
<
π
2
.
{\displaystyle {{K}_{\nu }}(z)=\int _{0}^{\infty }{{e}^{-z\cosh t}}\cosh(\nu t)dt,\qquad |Arg(z)|<{\frac {\pi }{2}}.}
K
ν
(
z
)
=
π
(
z
2
)
ν
Γ
(
ν
+
1
2
)
∫
1
∞
(
t
2
−
1
)
ν
−
1
2
e
−
z
t
d
t
,
R
e
(
ν
)
>
−
1
2
,
|
A
r
g
(
z
)
|
<
π
2
.
{\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
(
z
)
=
π
(
z
2
)
ν
Γ
(
ν
+
1
2
)
∫
0
∞
e
−
z
cosh
t
(
sinh
t
)
2
ν
d
t
,
R
e
(
ν
)
>
−
1
2
,
|
A
r
g
(
z
)
|
<
π
2
.
{\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}}.}
Asimptotski oblik
uredi
I
ν
(
z
)
∝
e
z
2
π
z
(
1
+
O
(
1
z
)
)
,
|
A
r
g
(
z
)
|
<
π
2
,
|
z
|
→
∞
.
{\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
(
z
)
∝
π
2
e
−
z
z
(
1
+
O
(
1
z
)
)
,
|
z
|
→
∞
.
{\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 .}
Literatura
uredi
![{\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)
