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Statements
I
μ
(
x
)
K
ν
(
x
)
=
∫
0
∞
J
μ
±
ν
(
2
x
sinh
t
)
e
(
-
μ
±
ν
)
t
d
t
,
modified-Bessel-first-kind
𝜇
𝑥
modified-Bessel-second-kind
𝜈
𝑥
superscript
subscript
0
Bessel-J
plus-or-minus
𝜇
𝜈
2
𝑥
𝑡
superscript
𝑒
plus-or-minus
𝜇
𝜈
𝑡
𝑡
{\displaystyle{\displaystyle I_{\mu}\left(x\right)K_{\nu}\left(x\right)=\int_{%
0}^{\infty}J_{\mu\pm\nu}\left(2x\sinh t\right)e^{(-\mu\pm\nu)t}\mathrm{d}t,}}
ℜ
(
μ
∓
ν
)
>
-
1
2
minus-or-plus
𝜇
𝜈
1
2
{\displaystyle{\displaystyle\Re(\mu\mp\nu)>-\tfrac{1}{2}}}
ℜ
(
μ
±
ν
)
>
-
1
plus-or-minus
𝜇
𝜈
1
{\displaystyle{\displaystyle\Re(\mu\pm\nu)>-1}}
ℜ
(
μ
∓
ν
)
>
-
1
2
minus-or-plus
𝜇
𝜈
1
2
{\displaystyle{\displaystyle\Re(\mu\mp\nu)>-\tfrac{1}{2}}}
ℜ
(
μ
±
ν
)
>
-
1
plus-or-minus
𝜇
𝜈
1
{\displaystyle{\displaystyle\Re(\mu\pm\nu)>-1}}
x
>
0
𝑥
0
{\displaystyle{\displaystyle x>0}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
sinh
z
𝑧
{\displaystyle{\displaystyle\sinh\NVar{z}}}
∫
{\displaystyle{\displaystyle\int}}
I
ν
(
z
)
modified-Bessel-first-kind
𝜈
𝑧
{\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
K
ν
(
z
)
modified-Bessel-second-kind
𝜈
𝑧
{\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
x
𝑥
{\displaystyle{\displaystyle x}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
