Функция
f
(
x
)
{\displaystyle f(x)}
Образ
F
(
τ
)
=
∫
0
∞
f
(
x
)
K
i
τ
(
x
)
d
x
,
τ
>
0.
{\displaystyle F(\tau )=\int _{0}^{\infty }f(x)K_{i\tau }(x)dx,\quad \tau >0.}
1
x
sin
(
α
x
)
,
|
I
m
α
|
<
π
2
{\displaystyle x\sin(\alpha x),\quad |\mathrm {Im} \,\alpha |<{\frac {\pi }{2}}}
π
τ
2
s
h
α
e
−
τ
c
h
α
{\displaystyle {\frac {\pi \tau }{2}}\mathrm {sh} \,\alpha e^{-\tau \mathrm {ch} \,\alpha }}
2
cos
(
α
x
)
,
|
I
m
α
|
<
π
2
{\displaystyle \cos(\alpha x),\quad |\mathrm {Im} \,\alpha |<{\frac {\pi }{2}}}
π
2
e
−
τ
c
h
α
{\displaystyle {\frac {\pi }{2}}e^{-\tau \mathrm {ch} \,\alpha }}
3
x
t
h
(
π
x
)
P
−
1
2
+
i
x
(
z
)
{\displaystyle x\mathrm {th} \,(\pi x)P_{-{\frac {1}{2}}+ix}(z)}
π
τ
2
e
−
τ
z
{\displaystyle {\sqrt {\frac {\pi \tau }{2}}}e^{-\tau z}}
4
x
t
h
(
π
x
)
K
i
x
(
z
)
,
|
a
r
g
z
|
<
π
{\displaystyle x\mathrm {th} \,(\pi x)K_{ix}(z),\quad |\mathrm {arg} \,z|<\pi }
π
2
τ
z
e
−
τ
−
z
z
+
τ
{\displaystyle {\frac {\pi }{2}}{\sqrt {\tau z}}{\frac {e^{-\tau -z}}{z+\tau }}}
5
x
s
h
(
π
x
)
K
2
i
x
(
z
)
,
|
a
r
g
z
|
<
π
4
{\displaystyle x\mathrm {sh} \,(\pi x)K_{2ix}(z),\quad |\mathrm {arg} \,z|<{\frac {\pi }{4}}}
π
3
z
2
2
5
τ
e
−
τ
−
z
2
8
τ
{\displaystyle {\sqrt {\frac {\pi ^{3}z^{2}}{2^{5}\tau }}}e^{-\tau -{\frac {z^{2}}{8\tau }}}}
6
x
sin
(
π
x
2
)
K
i
x
2
(
z
)
,
|
a
r
g
z
|
<
π
2
{\displaystyle x\sin({\frac {\pi x}{2}})K_{\frac {ix}{2}}(z),\quad |\mathrm {arg} \,z|<{\frac {\pi }{2}}}
π
3
τ
2
2
z
e
−
z
−
τ
2
8
z
{\displaystyle {\sqrt {\frac {\pi ^{3}\tau ^{2}}{2z}}}e^{-z-{\frac {\tau ^{2}}{8z}}}}
7
c
h
(
α
x
)
K
i
x
(
z
)
,
|
R
e
α
|
+
|
a
r
g
z
|
<
π
{\displaystyle \mathrm {ch} \,(\alpha x)K_{ix}(z),\quad |\mathrm {Re} \,\alpha |+|\mathrm {arg} \,z|<\pi }
π
2
K
0
(
τ
2
+
z
2
+
2
z
τ
cos
α
)
{\displaystyle {\frac {\pi }{2}}K_{0}({\sqrt {\tau ^{2}+z^{2}+2z\tau \cos \alpha }})}
8
x
x
2
+
n
2
s
h
(
π
x
)
K
i
x
(
z
)
,
z
>
0
,
n
∈
Z
+
{\displaystyle {\frac {x}{x^{2}+n^{2}}}\ \mathrm {sh} \,(\pi x)K_{ix}(z),\quad z>0,\ n\in \mathbb {Z} _{+}}
π
2
2
I
n
(
τ
)
K
n
(
z
)
,
0
<
τ
<
z
{\displaystyle {\frac {\pi ^{2}}{2}}I_{n}(\tau )K_{n}(z),0<\tau <z}
π
2
2
I
n
(
z
)
K
n
(
τ
)
,
z
<
τ
<
∞
{\displaystyle {\frac {\pi ^{2}}{2}}I_{n}(z)K_{n}(\tau ),z<\tau <\infty }
9
x
s
h
(
π
x
)
K
i
x
(
y
)
K
i
x
(
z
)
,
{\displaystyle x\mathrm {sh} \,(\pi x)K_{ix}(y)K_{ix}(z),}
|
a
r
g
y
|
+
|
a
r
g
z
|
<
π
2
{\displaystyle |\mathrm {arg} \,y|+|\mathrm {arg} \,z|<{\frac {\pi }{2}}}
π
2
4
e
−
τ
2
(
y
z
+
z
y
+
y
z
τ
2
)
{\displaystyle {\frac {\pi ^{2}}{4}}e^{-{\frac {\tau }{2}}\left({\frac {y}{z}}+{\frac {z}{y}}+{\frac {yz}{\tau ^{2}}}\right)}}
10
x
s
h
(
π
x
2
)
K
i
x
2
(
y
)
K
i
x
2
(
z
)
,
{\displaystyle x\mathrm {sh} \,({\frac {\pi x}{2}})K_{\frac {ix}{2}}(y)K_{\frac {ix}{2}}(z),}
|
a
r
g
y
|
+
|
a
r
g
z
|
<
π
{\displaystyle |\mathrm {arg} \,y|+|\mathrm {arg} \,z|<\pi }
π
2
τ
2
τ
2
+
4
y
z
e
−
(
y
+
z
)
2
τ
2
y
z
+
4
{\displaystyle {\frac {\pi ^{2}\tau }{2{\sqrt {\tau ^{2}+4yz}}}}e^{-{\frac {(y+z)}{2}}{\sqrt {{\frac {\tau ^{2}}{yz}}+4}}}}
11
x
s
h
(
π
x
)
K
i
x
2
+
λ
(
z
)
K
i
x
2
−
λ
(
z
)
,
z
>
0
{\displaystyle x\mathrm {sh} \,(\pi x)K_{{\frac {ix}{2}}+\lambda }(z)K_{{\frac {ix}{2}}-\lambda }(z),\quad z>0}
0
,
0
<
τ
<
2
z
{\displaystyle 0,0<\tau <2z}
π
2
τ
2
2
λ
+
1
z
2
λ
τ
2
−
4
z
2
(
(
τ
+
τ
2
−
4
z
2
)
2
λ
+
{\displaystyle {\frac {\pi ^{2}\tau }{2^{2\lambda +1}z^{2\lambda }{\sqrt {\tau ^{2}-4z^{2}}}}}\left((\tau +{\sqrt {\tau ^{2}-4z^{2}}})^{2\lambda }+\right.}
+
(
τ
−
τ
2
−
4
z
2
)
2
λ
)
,
2
z
<
τ
<
∞
{\displaystyle \left.+(\tau -{\sqrt {\tau ^{2}-4z^{2}}})^{2\lambda }\right),2z<\tau <\infty }
12
x
s
h
(
π
x
)
Γ
(
λ
+
i
x
)
Γ
(
λ
−
i
x
)
K
i
x
(
z
)
,
{\displaystyle x\mathrm {sh} \,(\pi x)\Gamma (\lambda +ix)\Gamma (\lambda -ix)K_{ix}(z),}
|
a
r
g
z
|
<
π
,
R
e
λ
>
0
{\displaystyle |\mathrm {arg} \,z|<\pi ,\;\mathrm {Re} \,\lambda >0}
2
λ
−
1
π
3
2
(
z
τ
)
λ
(
τ
+
z
)
−
λ
Γ
(
λ
+
1
2
)
K
λ
(
τ
+
z
)
{\displaystyle 2^{\lambda -1}\pi ^{\frac {3}{2}}(z\tau )^{\lambda }(\tau +z)^{-\lambda }\Gamma \left(\lambda +{\frac {1}{2}}\right)K_{\lambda }(\tau +z)}
