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Statements
∫
0
x
t
μ
J
ν
(
t
)
d
t
=
x
μ
Γ
(
1
2
ν
+
1
2
μ
+
1
2
)
Γ
(
1
2
ν
-
1
2
μ
+
1
2
)
∑
k
=
0
∞
(
ν
+
2
k
+
1
)
Γ
(
1
2
ν
-
1
2
μ
+
1
2
+
k
)
Γ
(
1
2
ν
+
1
2
μ
+
3
2
+
k
)
J
ν
+
2
k
+
1
(
x
)
,
superscript
subscript
0
𝑥
superscript
𝑡
𝜇
Bessel-J
𝜈
𝑡
𝑡
superscript
𝑥
𝜇
Euler-Gamma
1
2
𝜈
1
2
𝜇
1
2
Euler-Gamma
1
2
𝜈
1
2
𝜇
1
2
superscript
subscript
𝑘
0
𝜈
2
𝑘
1
Euler-Gamma
1
2
𝜈
1
2
𝜇
1
2
𝑘
Euler-Gamma
1
2
𝜈
1
2
𝜇
3
2
𝑘
Bessel-J
𝜈
2
𝑘
1
𝑥
{\displaystyle{\displaystyle\int_{0}^{x}t^{\mu}J_{\nu}\left(t\right)\mathrm{d}%
t=x^{\mu}\frac{\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{%
\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)}\*\sum_{k=0}^{%
\infty}\frac{(\nu+2k+1)\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k%
\right)}{\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k\right)}J_{\nu%
+2k+1}\left(x\right),}}
ℜ
(
μ
+
ν
+
1
)
>
0
𝜇
𝜈
1
0
{\displaystyle{\displaystyle\Re(\mu+\nu+1)>0}}
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)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
k
𝑘
{\displaystyle{\displaystyle k}}
x
𝑥
{\displaystyle{\displaystyle x}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
