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Statements
∫
0
∞
e
i
b
t
J
μ
(
a
t
)
d
t
=
{
exp
(
i
μ
arcsin
(
b
/
a
)
)
(
a
2
-
b
2
)
1
2
,
0
≤
b
<
a
,
i
a
μ
exp
(
1
2
μ
π
i
)
(
b
2
-
a
2
)
1
2
(
b
+
(
b
2
-
a
2
)
1
2
)
μ
,
0
<
a
<
b
.
superscript
subscript
0
superscript
𝑒
𝑖
𝑏
𝑡
Bessel-J
𝜇
𝑎
𝑡
𝑡
cases
𝑖
𝜇
𝑏
𝑎
superscript
superscript
𝑎
2
superscript
𝑏
2
1
2
0
𝑏
𝑎
𝑖
superscript
𝑎
𝜇
1
2
𝜇
𝜋
𝑖
superscript
superscript
𝑏
2
superscript
𝑎
2
1
2
superscript
𝑏
superscript
superscript
𝑏
2
superscript
𝑎
2
1
2
𝜇
0
𝑎
𝑏
{\displaystyle{\displaystyle\int_{0}^{\infty}e^{ibt}J_{\mu}\left(at\right)%
\mathrm{d}t=\begin{cases}\dfrac{\exp\left(i\mu\operatorname{arcsin}\left(b/a%
\right)\right)}{(a^{2}-b^{2})^{\frac{1}{2}}},&0\leq b<a,\\
\dfrac{ia^{\mu}\exp\left(\frac{1}{2}\mu\pi i\right)}{(b^{2}-a^{2})^{\frac{1}{2%
}}\left(b+(b^{2}-a^{2})^{\frac{1}{2}}\right)^{\mu}},&0<a<b.\end{cases}}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
exp
z
𝑧
{\displaystyle{\displaystyle\exp\NVar{z}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
arcsin
z
𝑧
{\displaystyle{\displaystyle\operatorname{arcsin}\NVar{z}}}
Identifiers
