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Statements
I
ν
(
z
)
=
(
1
2
z
)
ν
π
1
2
Γ
(
ν
+
1
2
)
∫
0
π
e
±
z
cos
θ
(
sin
θ
)
2
ν
d
θ
=
(
1
2
z
)
ν
π
1
2
Γ
(
ν
+
1
2
)
∫
-
1
1
(
1
-
t
2
)
ν
-
1
2
e
±
z
t
d
t
,
modified-Bessel-first-kind
𝜈
𝑧
superscript
1
2
𝑧
𝜈
superscript
𝜋
1
2
Euler-Gamma
𝜈
1
2
superscript
subscript
0
𝜋
superscript
𝑒
plus-or-minus
𝑧
𝜃
superscript
𝜃
2
𝜈
𝜃
superscript
1
2
𝑧
𝜈
superscript
𝜋
1
2
Euler-Gamma
𝜈
1
2
superscript
subscript
1
1
superscript
1
superscript
𝑡
2
𝜈
1
2
superscript
𝑒
plus-or-minus
𝑧
𝑡
𝑡
{\displaystyle{\displaystyle I_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}%
{\pi^{\frac{1}{2}}\Gamma\left(\nu+\frac{1}{2}\right)}\int_{0}^{\pi}e^{\pm z%
\cos\theta}(\sin\theta)^{2\nu}\mathrm{d}\theta=\frac{(\frac{1}{2}z)^{\nu}}{\pi%
^{\frac{1}{2}}\Gamma\left(\nu+\frac{1}{2}\right)}\int_{-1}^{1}(1-t^{2})^{\nu-%
\frac{1}{2}}e^{\pm zt}\mathrm{d}t,}}
ℜ
ν
>
-
1
2
𝜈
1
2
{\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
∫
{\displaystyle{\displaystyle\int}}
I
ν
(
z
)
modified-Bessel-first-kind
𝜈
𝑧
{\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
z
𝑧
{\displaystyle{\displaystyle z}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
