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Statements
J
ν
(
z
)
=
(
1
2
z
)
ν
π
1
2
Γ
(
ν
+
1
2
)
∫
0
π
cos
(
z
cos
θ
)
(
sin
θ
)
2
ν
d
θ
=
2
(
1
2
z
)
ν
π
1
2
Γ
(
ν
+
1
2
)
∫
0
1
(
1
-
t
2
)
ν
-
1
2
cos
(
z
t
)
d
t
,
Bessel-J
𝜈
𝑧
superscript
1
2
𝑧
𝜈
superscript
𝜋
1
2
Euler-Gamma
𝜈
1
2
superscript
subscript
0
𝜋
𝑧
𝜃
superscript
𝜃
2
𝜈
𝜃
2
superscript
1
2
𝑧
𝜈
superscript
𝜋
1
2
Euler-Gamma
𝜈
1
2
superscript
subscript
0
1
superscript
1
superscript
𝑡
2
𝜈
1
2
𝑧
𝑡
𝑡
{\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}%
}{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi}\cos\left%
(z\cos\theta\right)(\sin\theta)^{2\nu}\mathrm{d}\theta=\frac{2(\tfrac{1}{2}z)^%
{\nu}}{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{%
2})^{\nu-\frac{1}{2}}\cos\left(zt\right)\mathrm{d}t,}}
ℜ
ν
>
-
1
2
𝜈
1
2
{\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
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}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
z
𝑧
{\displaystyle{\displaystyle z}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
