Hiperboliskā sinusa sh , hiperboliskā kosinusa ch un hiperboliskā tangensa th grafiki Hiperboliskās funkcijas ir kompleksā vai reālā mainīgā analītiskās funkcijas . Tās ir analogās funkcijas trigonometriskajās funkcijām . Vienkāršākās hiperboliskās funkcijas ir hiperboliskais sinuss "sh" un hiperboliskais kosinuss "ch", no kuriem ir atvasināts hiperboliskais tangenss "th", hiperboliskais kosekanss "csch", hiperboliskais sekanss "sech" un hiperboliskais kotangenss "cth".
Hiperboliskās funkcijas parasti izmanto dažādu procesu (galvenokārt vienkāršu) raksturošanai, funkciju aproksimācijai .
sh
x
=
e
x
−
e
−
x
2
=
e
2
x
−
1
2
e
x
=
1
−
e
−
2
x
2
e
−
x
{\displaystyle \operatorname {sh} x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}}
ch
x
=
e
x
+
e
−
x
2
=
e
2
x
+
1
2
e
x
=
1
+
e
−
2
x
2
e
−
x
{\displaystyle \operatorname {ch} x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}}
th
x
=
sh
x
ch
x
=
e
x
−
e
−
x
e
x
+
e
−
x
=
e
2
x
−
1
e
2
x
+
1
=
1
−
e
−
2
x
1
+
e
−
2
x
{\displaystyle \operatorname {th} x={\frac {\operatorname {sh} x}{\operatorname {ch} x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}={\frac {1-e^{-2x}}{1+e^{-2x}}}}
Hiperboliskais kotangenss:
cth
x
=
ch
x
sh
x
=
e
x
+
e
−
x
e
x
−
e
−
x
=
e
2
x
+
1
e
2
x
−
1
=
1
+
e
−
2
x
1
−
e
−
2
x
{\displaystyle \operatorname {cth} x={\frac {\operatorname {ch} x}{\operatorname {sh} x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}={\frac {1+e^{-2x}}{1-e^{-2x}}}}
sech
x
=
(
ch
x
)
−
1
=
2
e
x
+
e
−
x
=
2
e
x
e
2
x
+
1
=
2
e
−
x
1
+
e
−
2
x
{\displaystyle \operatorname {sech} \,x=\left(\operatorname {ch} x\right)^{-1}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}={\frac {2e^{-x}}{1+e^{-2x}}}}
Hiperboliskais kosekanss:
csch
x
=
(
sh
x
)
−
1
=
2
e
x
−
e
−
x
=
2
e
x
e
2
x
−
1
=
2
e
−
x
1
−
e
−
2
x
{\displaystyle \operatorname {csch} \,x=\left(\operatorname {sh} x\right)^{-1}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}={\frac {2e^{-x}}{1-e^{-2x}}}}
Hiperboliskās funkcijas var izteikt arī ar trigonometriskajām funkcijām :
sh
x
=
−
i
sin
i
x
{\displaystyle \operatorname {sh} x=-{\rm {i}}\sin {\rm {i}}x\!}
ch
x
=
cos
i
x
{\displaystyle \operatorname {ch} x=\cos {\rm {i}}x\!}
th
x
=
−
i
tg
i
x
{\displaystyle \operatorname {th} x=-{\rm {i}}\operatorname {tg} {\rm {i}}x\!}
Hiperboliskais kotangenss:
cth
x
=
i
ctg
i
x
{\displaystyle \operatorname {cth} x={\rm {i}}\operatorname {ctg} {\rm {i}}x\!}
sech
x
=
sec
i
x
{\displaystyle \operatorname {sech} \,x=\sec {{\rm {i}}x}\!}
Hiperboliskais kosekanss:
csch
x
=
i
csc
i
x
{\displaystyle \operatorname {csch} \,x={\rm {i}}\,\csc \,{\rm {i}}x\!}
kur i ir imaginārā vienība : i 2 = −1.
Pāra un nepāra funkcijas:
sh
−
x
=
−
sh
x
ch
−
x
=
ch
x
{\displaystyle {\begin{aligned}\operatorname {sh} -x&=-\operatorname {sh} x\\\operatorname {ch} -x&=\operatorname {ch} x\end{aligned}}}
Tātad:
th
−
x
=
−
th
x
cth
−
x
=
−
cth
x
sech
−
x
=
sech
x
csch
−
x
=
−
csch
x
{\displaystyle {\begin{aligned}\operatorname {th} -x&=-\operatorname {th} x\\\operatorname {cth} -x&=-\operatorname {cth} x\\\operatorname {sech} -x&=\operatorname {sech} x\\\operatorname {csch} -x&=-\operatorname {csch} x\end{aligned}}}
Hiperboliskais sinuss un hiperboliskais kosinuss apmierina vienādību
ch
2
x
−
sh
2
x
=
1
{\displaystyle \operatorname {ch} ^{2}x-\operatorname {sh} ^{2}x=1\,}
Inversās hiperboliskās trigonometriskās funkcijas var izteikt ar naturāllogaritmiem
arsh
(
x
)
=
ln
(
x
+
x
2
+
1
)
arch
(
x
)
=
ln
(
x
+
x
2
−
1
)
;
x
≥
1
arth
(
x
)
=
1
2
ln
(
1
+
x
1
−
x
)
;
|
x
|
<
1
arcth
(
x
)
=
1
2
ln
(
x
+
1
x
−
1
)
;
|
x
|
>
1
arsech
(
x
)
=
ln
(
1
x
+
1
−
x
2
x
)
;
0
<
x
≤
1
arcsch
(
x
)
=
ln
(
1
x
+
1
+
x
2
|
x
|
)
;
x
≠
0
{\displaystyle {\begin{aligned}\operatorname {arsh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arch} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right);x\geq 1\\\operatorname {arth} (x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right);\left|x\right|<1\\\operatorname {arcth} (x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right);\left|x\right|>1\\\operatorname {arsech} (x)&=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1-x^{2}}}{x}}\right);0<x\leq 1\\\operatorname {arcsch} (x)&=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1+x^{2}}}{\left|x\right|}}\right);x\neq 0\end{aligned}}}
d
d
x
sh
x
=
ch
x
{\displaystyle {\frac {d}{dx}}\operatorname {sh} x=\operatorname {ch} x\,}
d
d
x
ch
x
=
sh
x
{\displaystyle {\frac {d}{dx}}\operatorname {ch} x=\operatorname {sh} x\,}
d
d
x
th
x
=
1
−
th
2
x
=
sech
2
x
=
1
/
ch
2
x
{\displaystyle {\frac {d}{dx}}\operatorname {th} x=1-\operatorname {th} ^{2}x=\operatorname {sech} ^{2}x=1/\operatorname {ch} ^{2}x\,}
d
d
x
cth
x
=
1
−
cth
2
x
=
−
csch
2
x
=
−
1
/
sh
2
x
{\displaystyle {\frac {d}{dx}}\operatorname {cth} x=1-\operatorname {cth} ^{2}x=-\operatorname {csch} ^{2}x=-1/\operatorname {sh} ^{2}x\,}
d
d
x
csch
x
=
−
cth
x
csch
x
{\displaystyle {\frac {d}{dx}}\ \operatorname {csch} \,x=-\operatorname {cth} x\ \operatorname {csch} \,x\,}
d
d
x
sech
x
=
−
th
x
sech
x
{\displaystyle {\frac {d}{dx}}\ \operatorname {sech} \,x=-\operatorname {th} x\ \operatorname {sech} \,x\,}
d
d
x
arsh
x
=
1
x
2
+
1
{\displaystyle {\frac {d}{dx}}\,\operatorname {arsh} \,x={\frac {1}{\sqrt {x^{2}+1}}}}
d
d
x
arch
x
=
1
x
2
−
1
{\displaystyle {\frac {d}{dx}}\,\operatorname {arch} \,x={\frac {1}{\sqrt {x^{2}-1}}}}
d
d
x
arth
x
=
1
1
−
x
2
{\displaystyle {\frac {d}{dx}}\,\operatorname {arth} \,x={\frac {1}{1-x^{2}}}}
d
d
x
arcsch
x
=
−
1
|
x
|
1
+
x
2
{\displaystyle {\frac {d}{dx}}\,\operatorname {arcsch} \,x=-{\frac {1}{\left|x\right|{\sqrt {1+x^{2}}}}}}
d
d
x
arsech
x
=
−
1
x
1
−
x
2
{\displaystyle {\frac {d}{dx}}\,\operatorname {arsech} \,x=-{\frac {1}{x{\sqrt {1-x^{2}}}}}}
d
d
x
arcth
x
=
1
1
−
x
2
{\displaystyle {\frac {d}{dx}}\,\operatorname {arcth} \,x={\frac {1}{1-x^{2}}}}
∫
sh
(
a
x
)
d
x
=
a
−
1
ch
(
a
x
)
+
C
∫
ch
(
a
x
)
d
x
=
a
−
1
sh
(
a
x
)
+
C
∫
th
(
a
x
)
d
x
=
a
−
1
ln
(
ch
(
a
x
)
)
+
C
∫
cth
(
a
x
)
d
x
=
a
−
1
ln
(
sh
(
a
x
)
)
+
C
∫
sech
(
a
x
)
d
x
=
a
−
1
arctg
(
sh
(
a
x
)
)
+
C
∫
csch
(
a
x
)
d
x
=
a
−
1
ln
(
th
(
a
x
2
)
)
+
C
{\displaystyle {\begin{aligned}\int \operatorname {sh} (ax)\,dx&=a^{-1}\operatorname {ch} (ax)+C\\\int \operatorname {ch} (ax)\,dx&=a^{-1}\operatorname {sh} (ax)+C\\\int \operatorname {th} (ax)\,dx&=a^{-1}\ln(\operatorname {ch} (ax))+C\\\int \operatorname {cth} (ax)\,dx&=a^{-1}\ln(\operatorname {sh} (ax))+C\\\int \operatorname {sech} (ax)\,dx&=a^{-1}\operatorname {arctg} (\operatorname {sh} (ax))+C\\\int \operatorname {csch} (ax)\,dx&=a^{-1}\ln \left(\operatorname {th} \left({\frac {ax}{2}}\right)\right)+C\end{aligned}}}
∫
d
u
a
2
+
u
2
=
sh
−
1
(
u
a
)
+
C
∫
d
u
u
2
−
a
2
=
ch
−
1
(
u
a
)
+
C
∫
d
u
a
2
−
u
2
=
a
−
1
th
−
1
(
u
a
)
+
C
;
u
2
<
a
2
∫
d
u
a
2
−
u
2
=
a
−
1
cth
−
1
(
u
a
)
+
C
;
u
2
>
a
2
∫
d
u
u
a
2
−
u
2
=
−
a
−
1
sech
−
1
(
u
a
)
+
C
∫
d
u
u
a
2
+
u
2
=
−
a
−
1
csch
−
1
|
u
a
|
+
C
{\displaystyle {\begin{aligned}\int {\frac {du}{\sqrt {a^{2}+u^{2}}}}&=\operatorname {sh} ^{-1}\left({\frac {u}{a}}\right)+C\\\int {\frac {du}{\sqrt {u^{2}-a^{2}}}}&=\operatorname {ch} ^{-1}\left({\frac {u}{a}}\right)+C\\\int {\frac {du}{a^{2}-u^{2}}}&=a^{-1}\operatorname {th} ^{-1}\left({\frac {u}{a}}\right)+C;u^{2}<a^{2}\\\int {\frac {du}{a^{2}-u^{2}}}&=a^{-1}\operatorname {cth} ^{-1}\left({\frac {u}{a}}\right)+C;u^{2}>a^{2}\\\int {\frac {du}{u{\sqrt {a^{2}-u^{2}}}}}&=-a^{-1}\operatorname {sech} ^{-1}\left({\frac {u}{a}}\right)+C\\\int {\frac {du}{u{\sqrt {a^{2}+u^{2}}}}}&=-a^{-1}\operatorname {csch} ^{-1}\left|{\frac {u}{a}}\right|+C\end{aligned}}}
kur C ir integrēšanas konstante.
Funkciju izvirzījumi Teilora rindā :
sh
x
=
x
+
x
3
3
!
+
x
5
5
!
+
x
7
7
!
+
⋯
=
∑
n
=
0
∞
x
2
n
+
1
(
2
n
+
1
)
!
{\displaystyle \operatorname {sh} x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}}
ch
x
=
1
+
x
2
2
!
+
x
4
4
!
+
x
6
6
!
+
⋯
=
∑
n
=
0
∞
x
2
n
(
2
n
)
!
{\displaystyle \operatorname {ch} x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}}
th
x
=
x
−
x
3
3
+
2
x
5
15
−
17
x
7
315
+
⋯
=
∑
n
=
1
∞
2
2
n
(
2
2
n
−
1
)
B
2
n
x
2
n
−
1
(
2
n
)
!
,
|
x
|
<
π
2
cth
x
=
x
−
1
+
x
3
−
x
3
45
+
2
x
5
945
+
⋯
=
x
−
1
+
∑
n
=
1
∞
2
2
n
B
2
n
x
2
n
−
1
(
2
n
)
!
,
0
<
|
x
|
<
π
sech
x
=
1
−
x
2
2
+
5
x
4
24
−
61
x
6
720
+
⋯
=
∑
n
=
0
∞
E
2
n
x
2
n
(
2
n
)
!
,
|
x
|
<
π
2
csch
x
=
x
−
1
−
x
6
+
7
x
3
360
−
31
x
5
15120
+
⋯
=
x
−
1
+
∑
n
=
1
∞
2
(
1
−
2
2
n
−
1
)
B
2
n
x
2
n
−
1
(
2
n
)
!
,
0
<
|
x
|
<
π
{\displaystyle {\begin{aligned}\operatorname {th} x&=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}\\\operatorname {cth} x&=x^{-1}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\cdots =x^{-1}+\sum _{n=1}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi \\\operatorname {sech} \,x&=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\cdots =\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}\\\operatorname {csch} \,x&=x^{-1}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\cdots =x^{-1}+\sum _{n=1}^{\infty }{\frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi \end{aligned}}}
where
B
n
{\displaystyle B_{n}\,}
n -tais Bernulli skaitlis
E
n
{\displaystyle E_{n}\,}
n -tais Eilera skaitlis 
