Sò͘-ha̍k siōng, siang-khiok hâm-sò͘ (Hàn-jī : 雙曲函數 saⁿ-kak hâm-sò͘ ê lūi-chhui, m̄-koh in sī ēng siang-khiok-sòaⁿ tēng-gī ·ê, m̄ sī ēng îⁿ-hêng tēng-gī ·ê. Tiō chhin-chhiūⁿ (cos t , sin t ) chia-ê tiám hêng-sêng chi̍t ê tan-ūi-îⁿ kāng-khoán, (cosh t , sinh t ) chia-ê tiám ē tan-ūi siang-khiok-sòaⁿ ê chiàⁿ-chhiú pòaⁿ-pêng. Koh ū chi̍t ê sio-siâng ê só͘-chāi sī, sin(t ) kap cos(t ) ê tō-hâm-sò͘ hun-pia̍t sī cos(t ) kap –sin(t ) , ah sinh(t ) kap cosh(t ) ê tō-hâm-só͘ hun-pia̍t sī cosh(t ) kap +sinh(t ) .
Siang-khiok hâm-sò͘ chhut-hiān tī siang-khiok kì-hō-ha̍k tiong kak-tō͘ kap kī-lī ê kè-sǹg. They also occur in the solutions of many sòaⁿ-sèng bî-hun hong-têng-sek (phì-lūn-kóng tēng-gī catenary ê hong-têng-sek), li̍p-hong hong-têng-sek, kap Khu-chhioh-kak chō-piau tiong ê Laplace hong-têng-sek . Laplace hong-têng-sek tī bu̍t-lí-ha̍k ê chin-chē léng-he̍k tiong lóng chiok tiōng-iàu ·ê, chia-ê léng-he̍k pau-koat tiān-chú-ha̍k , jia̍t thoân-tō , liû-thé le̍k-ha̍k kap te̍k-sû siong-tùi-lūn .
Ki-pún ê siang-khiok hâm-sò͘ ū:[ 1]
siang-khiok sine "sinh " ([ 2] siang-khiok cosine "cosh " ([ 3] tùi chit nn̄g ê ē-sái tit-tio̍h:[ 4]
siang-khiok tangent "tanh " ([ 5] siang-khiok cosecant "csch " or "cosech " ([ 6] siang-khiok secant "sech " ([ 7] siang-khiok cotangent "coth " ([ 8] [ 9] Beh tēng-gī siang-khiok hâm-sò͘, ū chiok chē chióng hoat.
sinh x sī ex e −x chha ê chi̍t-pòaⁿcosh x sī ex e −x pêng-kun Tùi chí-sò͘ hâm-sò͘ lâi tēng-gī:[ 10] [ 11]
Siang-khiok sine: chí-sò͘ hâm-sò͘ ê khia-pō͘ , tiō-sī,
sinh
x
=
e
x
−
e
−
x
2
=
e
2
x
−
1
2
e
x
=
1
−
e
−
2
x
2
e
−
x
.
{\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.}
Siang-khiok cosine: the even part of the exponential function, that is,
cosh
x
=
e
x
+
e
−
x
2
=
e
2
x
+
1
2
e
x
=
1
+
e
−
2
x
2
e
−
x
.
{\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.}
Siang-khiok tangent:
tanh
x
=
sinh
x
cosh
x
=
e
x
−
e
−
x
e
x
+
e
−
x
=
e
2
x
−
1
e
2
x
+
1
.
{\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}.}
Siang-khiok cotangent: x ≠ 0
coth
x
=
cosh
x
sinh
x
=
e
x
+
e
−
x
e
x
−
e
−
x
=
e
2
x
+
1
e
2
x
−
1
.
{\displaystyle \coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}.}
Siang-khiok secant:
sech
x
=
1
cosh
x
=
2
e
x
+
e
−
x
=
2
e
x
e
2
x
+
1
.
{\displaystyle \operatorname {sech} x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}.}
Siang-khiok cosecant: x ≠ 0
csch
x
=
1
sinh
x
=
2
e
x
−
e
−
x
=
2
e
x
e
2
x
−
1
.
{\displaystyle \operatorname {csch} x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}.}
Tùi ho̍k-cha̍p-sò͘ piān-só͘ ê saⁿ-kak hâm-sò͘ mā ē-sái lâi tēng-gī siang-khiok hâm-sò͘:
Siang-khiok sine:[ 12]
sinh
x
=
−
i
sin
(
i
x
)
.
{\displaystyle \sinh x=-i\sin(ix).}
Siang-khiok cosine:[ 13]
cosh
x
=
cos
(
i
x
)
.
{\displaystyle \cosh x=\cos(ix).}
Siang-khiok tangent:
tanh
x
=
−
i
tan
(
i
x
)
.
{\displaystyle \tanh x=-i\tan(ix).}
Siang-khiok cotangent:
coth
x
=
i
cot
(
i
x
)
.
{\displaystyle \coth x=i\cot(ix).}
Siang-khiok secant:
sech
x
=
sec
(
i
x
)
.
{\displaystyle \operatorname {sech} x=\sec(ix).}
Siang-khiok cosecant:
csch
x
=
i
csc
(
i
x
)
.
{\displaystyle \operatorname {csch} x=i\csc(ix).}
kî-tiong i sī hi-sò͘ tan-ūi-goân , i 2 = −1
Í-siōng ê tēng-gī sī thàu-kòe Euler kong-sek kap chí-sò͘ hâm-sò͘ tēng-gī sán-seng koan-hē.
sinh kap cosh hun-pia̍t sī khia-sò͘ hâm-sò͘ kap siang-sò͘ hâm-sò͘:
sinh
(
−
x
)
=
−
sinh
x
cosh
(
−
x
)
=
cosh
x
{\displaystyle {\begin{aligned}\sinh(-x)&=-\sinh x\\\cosh(-x)&=\cosh x\end{aligned}}}
tanh
(
−
x
)
=
−
tanh
x
coth
(
−
x
)
=
−
coth
x
sech
(
−
x
)
=
sech
x
csch
(
−
x
)
=
−
csch
x
{\displaystyle {\begin{aligned}\tanh(-x)&=-\tanh x\\\coth(-x)&=-\coth x\\\operatorname {sech} (-x)&=\operatorname {sech} x\\\operatorname {csch} (-x)&=-\operatorname {csch} x\end{aligned}}}
cosh x kap sech x sī siang-sò͘ hâm-sò͘ ; kî-thaⁿ-·ê sī khia-sò͘ hâm-sò͘ .
arsech
x
=
arcosh
(
1
x
)
arcsch
x
=
arsinh
(
1
x
)
arcoth
x
=
artanh
(
1
x
)
{\displaystyle {\begin{aligned}\operatorname {arsech} x&=\operatorname {arcosh} \left({\frac {1}{x}}\right)\\\operatorname {arcsch} x&=\operatorname {arsinh} \left({\frac {1}{x}}\right)\\\operatorname {arcoth} x&=\operatorname {artanh} \left({\frac {1}{x}}\right)\end{aligned}}}
cosh
x
+
sinh
x
=
e
x
cosh
x
−
sinh
x
=
e
−
x
cosh
2
x
−
sinh
2
x
=
1
{\displaystyle {\begin{aligned}\cosh x+\sinh x&=e^{x}\\\cosh x-\sinh x&=e^{-x}\\\cosh ^{2}x-\sinh ^{2}x&=1\end{aligned}}}
Pythagorean saⁿ-kak hêng-têng-sek ū sêng.
Lán mā ū
sech
2
x
=
1
−
tanh
2
x
csch
2
x
=
coth
2
x
−
1
{\displaystyle {\begin{aligned}\operatorname {sech} ^{2}x&=1-\tanh ^{2}x\\\operatorname {csch} ^{2}x&=\coth ^{2}x-1\end{aligned}}}
sinh
(
x
+
y
)
=
sinh
x
cosh
y
+
cosh
x
sinh
y
cosh
(
x
+
y
)
=
cosh
x
cosh
y
+
sinh
x
sinh
y
tanh
(
x
+
y
)
=
tanh
x
+
tanh
y
1
+
tanh
x
tanh
y
{\displaystyle {\begin{aligned}\sinh(x+y)&=\sinh x\cosh y+\cosh x\sinh y\\\cosh(x+y)&=\cosh x\cosh y+\sinh x\sinh y\\[6px]\tanh(x+y)&={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\\\end{aligned}}}
cosh
(
2
x
)
=
sinh
2
x
+
cosh
2
x
=
2
sinh
2
x
+
1
=
2
cosh
2
x
−
1
sinh
(
2
x
)
=
2
sinh
x
cosh
x
tanh
(
2
x
)
=
2
tanh
x
1
+
tanh
2
x
{\displaystyle {\begin{aligned}\cosh(2x)&=\sinh ^{2}{x}+\cosh ^{2}{x}=2\sinh ^{2}x+1=2\cosh ^{2}x-1\\\sinh(2x)&=2\sinh x\cosh x\\\tanh(2x)&={\frac {2\tanh x}{1+\tanh ^{2}x}}\\\end{aligned}}}
sinh
x
+
sinh
y
=
2
sinh
(
x
+
y
2
)
cosh
(
x
−
y
2
)
cosh
x
+
cosh
y
=
2
cosh
(
x
+
y
2
)
cosh
(
x
−
y
2
)
{\displaystyle {\begin{aligned}\sinh x+\sinh y&=2\sinh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\cosh x+\cosh y&=2\cosh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\end{aligned}}}
sinh
(
x
−
y
)
=
sinh
x
cosh
y
−
cosh
x
sinh
y
cosh
(
x
−
y
)
=
cosh
x
cosh
y
−
sinh
x
sinh
y
tanh
(
x
−
y
)
=
tanh
x
−
tanh
y
1
−
tanh
x
tanh
y
{\displaystyle {\begin{aligned}\sinh(x-y)&=\sinh x\cosh y-\cosh x\sinh y\\\cosh(x-y)&=\cosh x\cosh y-\sinh x\sinh y\\\tanh(x-y)&={\frac {\tanh x-\tanh y}{1-\tanh x\tanh y}}\\\end{aligned}}}
[ 14]
sinh
x
−
sinh
y
=
2
cosh
(
x
+
y
2
)
sinh
(
x
−
y
2
)
cosh
x
−
cosh
y
=
2
sinh
(
x
+
y
2
)
sinh
(
x
−
y
2
)
{\displaystyle {\begin{aligned}\sinh x-\sinh y&=2\cosh \left({\frac {x+y}{2}}\right)\sinh \left({\frac {x-y}{2}}\right)\\\cosh x-\cosh y&=2\sinh \left({\frac {x+y}{2}}\right)\sinh \left({\frac {x-y}{2}}\right)\\\end{aligned}}}
sinh
(
x
2
)
=
sinh
x
2
(
cosh
x
+
1
)
=
sgn
x
cosh
x
−
1
2
cosh
(
x
2
)
=
cosh
x
+
1
2
tanh
(
x
2
)
=
sinh
x
cosh
x
+
1
=
sgn
x
cosh
x
−
1
cosh
x
+
1
=
e
x
−
1
e
x
+
1
{\displaystyle {\begin{aligned}\sinh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\sqrt {2(\cosh x+1)}}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{2}}}\\[6px]\cosh \left({\frac {x}{2}}\right)&={\sqrt {\frac {\cosh x+1}{2}}}\\[6px]\tanh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\cosh x+1}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{\cosh x+1}}}={\frac {e^{x}-1}{e^{x}+1}}\end{aligned}}}
sgn sī chèng-hū-hō hâm-sò͘ .[ 15]
x ≠ 0
tanh
(
x
2
)
=
cosh
x
−
1
sinh
x
=
coth
x
−
csch
x
{\displaystyle \tanh \left({\frac {x}{2}}\right)={\frac {\cosh x-1}{\sinh x}}=\coth x-\operatorname {csch} x}
sinh
2
x
=
1
2
(
cosh
2
x
−
1
)
cosh
2
x
=
1
2
(
cosh
2
x
+
1
)
{\displaystyle {\begin{aligned}\sinh ^{2}x&={\frac {1}{2}}(\cosh 2x-1)\\\cosh ^{2}x&={\frac {1}{2}}(\cosh 2x+1)\end{aligned}}}
Chit ê put-téng-sek tī thóng-kè-ha̍k lāi-té chin ū lō͘-ēng:
cosh
(
t
)
≤
e
t
2
/
2
{\displaystyle \operatorname {cosh} (t)\leq e^{t^{2}/2}}
[ 16]
arsinh
(
x
)
=
ln
(
x
+
x
2
+
1
)
arcosh
(
x
)
=
ln
(
x
+
x
2
−
1
)
x
≥
1
artanh
(
x
)
=
1
2
ln
(
1
+
x
1
−
x
)
|
x
|
<
1
arcoth
(
x
)
=
1
2
ln
(
x
+
1
x
−
1
)
|
x
|
>
1
arsech
(
x
)
=
ln
(
1
x
+
1
x
2
−
1
)
=
ln
(
1
+
1
−
x
2
x
)
0
<
x
≤
1
arcsch
(
x
)
=
ln
(
1
x
+
1
x
2
+
1
)
x
≠
0
{\displaystyle {\begin{aligned}\operatorname {arsinh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right)&&x\geq 1\\\operatorname {artanh} (x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)&&|x|<1\\\operatorname {arcoth} (x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)&&|x|>1\\\operatorname {arsech} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}-1}}\right)=\ln \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)&&0<x\leq 1\\\operatorname {arcsch} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}+1}}\right)&&x\neq 0\end{aligned}}}
d
d
x
sinh
x
=
cosh
x
d
d
x
cosh
x
=
sinh
x
d
d
x
tanh
x
=
1
−
tanh
2
x
=
sech
2
x
=
1
cosh
2
x
d
d
x
coth
x
=
1
−
coth
2
x
=
−
csch
2
x
=
−
1
sinh
2
x
x
≠
0
d
d
x
sech
x
=
−
tanh
x
sech
x
d
d
x
csch
x
=
−
coth
x
csch
x
x
≠
0
{\displaystyle {\begin{aligned}{\frac {d}{dx}}\sinh x&=\cosh x\\{\frac {d}{dx}}\cosh x&=\sinh x\\{\frac {d}{dx}}\tanh x&=1-\tanh ^{2}x=\operatorname {sech} ^{2}x={\frac {1}{\cosh ^{2}x}}\\{\frac {d}{dx}}\coth x&=1-\coth ^{2}x=-\operatorname {csch} ^{2}x=-{\frac {1}{\sinh ^{2}x}}&&x\neq 0\\{\frac {d}{dx}}\operatorname {sech} x&=-\tanh x\operatorname {sech} x\\{\frac {d}{dx}}\operatorname {csch} x&=-\coth x\operatorname {csch} x&&x\neq 0\end{aligned}}}
d
d
x
arsinh
x
=
1
x
2
+
1
d
d
x
arcosh
x
=
1
x
2
−
1
1
<
x
d
d
x
artanh
x
=
1
1
−
x
2
|
x
|
<
1
d
d
x
arcoth
x
=
1
1
−
x
2
1
<
|
x
|
d
d
x
arsech
x
=
−
1
x
1
−
x
2
0
<
x
<
1
d
d
x
arcsch
x
=
−
1
|
x
|
1
+
x
2
x
≠
0
{\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arsinh} x&={\frac {1}{\sqrt {x^{2}+1}}}\\{\frac {d}{dx}}\operatorname {arcosh} x&={\frac {1}{\sqrt {x^{2}-1}}}&&1<x\\{\frac {d}{dx}}\operatorname {artanh} x&={\frac {1}{1-x^{2}}}&&|x|<1\\{\frac {d}{dx}}\operatorname {arcoth} x&={\frac {1}{1-x^{2}}}&&1<|x|\\{\frac {d}{dx}}\operatorname {arsech} x&=-{\frac {1}{x{\sqrt {1-x^{2}}}}}&&0<x<1\\{\frac {d}{dx}}\operatorname {arcsch} x&=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}&&x\neq 0\end{aligned}}}
sinh kap cosh ê jī-chhù tō-hâm-sò͘ tiō sī in ka-kī:
d
2
d
x
2
sinh
x
=
sinh
x
{\displaystyle {\frac {d^{2}}{dx^{2}}}\sinh x=\sinh x}
d
2
d
x
2
cosh
x
=
cosh
x
.
{\displaystyle {\frac {d^{2}}{dx^{2}}}\cosh x=\cosh x\,.}
sinh kap cosh ê sòaⁿ-sèng cho͘-ha̍p, te̍k-pia̍t pau-koat chí-sò͘ hâm-sò͘
e
x
{\displaystyle e^{x}}
e
−
x
{\displaystyle e^{-x}}
∫
sinh
(
a
x
)
d
x
=
a
−
1
cosh
(
a
x
)
+
C
∫
cosh
(
a
x
)
d
x
=
a
−
1
sinh
(
a
x
)
+
C
∫
tanh
(
a
x
)
d
x
=
a
−
1
ln
(
cosh
(
a
x
)
)
+
C
∫
coth
(
a
x
)
d
x
=
a
−
1
ln
|
sinh
(
a
x
)
|
+
C
∫
sech
(
a
x
)
d
x
=
a
−
1
arctan
(
sinh
(
a
x
)
)
+
C
∫
csch
(
a
x
)
d
x
=
a
−
1
ln
|
tanh
(
a
x
2
)
|
+
C
=
a
−
1
ln
|
coth
(
a
x
)
−
csch
(
a
x
)
|
+
C
=
−
a
−
1
arcoth
(
cosh
(
a
x
)
)
+
C
{\displaystyle {\begin{aligned}\int \sinh(ax)\,dx&=a^{-1}\cosh(ax)+C\\\int \cosh(ax)\,dx&=a^{-1}\sinh(ax)+C\\\int \tanh(ax)\,dx&=a^{-1}\ln(\cosh(ax))+C\\\int \coth(ax)\,dx&=a^{-1}\ln \left|\sinh(ax)\right|+C\\\int \operatorname {sech} (ax)\,dx&=a^{-1}\arctan(\sinh(ax))+C\\\int \operatorname {csch} (ax)\,dx&=a^{-1}\ln \left|\tanh \left({\frac {ax}{2}}\right)\right|+C=a^{-1}\ln \left|\coth \left(ax\right)-\operatorname {csch} \left(ax\right)\right|+C=-a^{-1}\operatorname {arcoth} \left(\cosh \left(ax\right)\right)+C\end{aligned}}}
∫
1
a
2
+
u
2
d
u
=
arsinh
(
u
a
)
+
C
∫
1
u
2
−
a
2
d
u
=
sgn
u
arcosh
|
u
a
|
+
C
∫
1
a
2
−
u
2
d
u
=
a
−
1
artanh
(
u
a
)
+
C
u
2
<
a
2
∫
1
a
2
−
u
2
d
u
=
a
−
1
arcoth
(
u
a
)
+
C
u
2
>
a
2
∫
1
u
a
2
−
u
2
d
u
=
−
a
−
1
arsech
|
u
a
|
+
C
∫
1
u
a
2
+
u
2
d
u
=
−
a
−
1
arcsch
|
u
a
|
+
C
{\displaystyle {\begin{aligned}\int {{\frac {1}{\sqrt {a^{2}+u^{2}}}}\,du}&=\operatorname {arsinh} \left({\frac {u}{a}}\right)+C\\\int {{\frac {1}{\sqrt {u^{2}-a^{2}}}}\,du}&=\operatorname {sgn} {u}\operatorname {arcosh} \left|{\frac {u}{a}}\right|+C\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {artanh} \left({\frac {u}{a}}\right)+C&&u^{2}<a^{2}\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {arcoth} \left({\frac {u}{a}}\right)+C&&u^{2}>a^{2}\\\int {{\frac {1}{u{\sqrt {a^{2}-u^{2}}}}}\,du}&=-a^{-1}\operatorname {arsech} \left|{\frac {u}{a}}\right|+C\\\int {{\frac {1}{u{\sqrt {a^{2}+u^{2}}}}}\,du}&=-a^{-1}\operatorname {arcsch} \left|{\frac {u}{a}}\right|+C\end{aligned}}}
C sī chek-hun siông-sò͘ .
Kā chí-sò͘ hâm-sò͘ hun-kái chò i ê khia-sò͘ pō͘-hūn kap siang-sò͘ pō͘-hūn , lán tiō ū hêng-téng-sek
e
x
=
cosh
x
+
sinh
x
,
{\displaystyle e^{x}=\cosh x+\sinh x,}
e
−
x
=
cosh
x
−
sinh
x
.
{\displaystyle e^{-x}=\cosh x-\sinh x.}
e
i
x
=
cos
x
+
i
sin
x
,
{\displaystyle e^{ix}=\cos x+i\sin x,}
e
x
+
i
y
=
(
cosh
x
+
sinh
x
)
(
cos
y
+
i
sin
y
)
{\displaystyle e^{x+iy}=(\cosh x+\sinh x)(\cos y+i\sin y)}
it-poaⁿ ê ho̍k-cha̍p-sò͘ chí-sò͘ hâm-sò͘ ê kong-sek.
Lēng-gōa koh ū
e
x
=
1
+
tanh
x
1
−
tanh
x
=
1
+
tanh
x
2
1
−
tanh
x
2
{\displaystyle e^{x}={\sqrt {\frac {1+\tanh x}{1-\tanh x}}}={\frac {1+\tanh {\frac {x}{2}}}{1-\tanh {\frac {x}{2}}}}}
In-ūi chí-sò͘ hâm-sò͘ ē-sái tùi jīm-hô ho̍k-cha̍p-sò͘ piān-só͘ lâi tēng-gī, lán tiō ē-sái kā hit khoán tēng-gī chhun kàu siang-khiok hâm-sò͘.
Ho̍k-cha̍p-sò͘ ê Euler kong-sek
e
i
x
=
cos
x
+
i
sin
x
e
−
i
x
=
cos
x
−
i
sin
x
{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\e^{-ix}&=\cos x-i\sin x\end{aligned}}}
cosh
(
i
x
)
=
1
2
(
e
i
x
+
e
−
i
x
)
=
cos
x
sinh
(
i
x
)
=
1
2
(
e
i
x
−
e
−
i
x
)
=
i
sin
x
cosh
(
x
+
i
y
)
=
cosh
(
x
)
cos
(
y
)
+
i
sinh
(
x
)
sin
(
y
)
sinh
(
x
+
i
y
)
=
sinh
(
x
)
cos
(
y
)
+
i
cosh
(
x
)
sin
(
y
)
tanh
(
i
x
)
=
i
tan
x
cosh
x
=
cos
(
i
x
)
sinh
x
=
−
i
sin
(
i
x
)
tanh
x
=
−
i
tan
(
i
x
)
{\displaystyle {\begin{aligned}\cosh(ix)&={\frac {1}{2}}\left(e^{ix}+e^{-ix}\right)=\cos x\\\sinh(ix)&={\frac {1}{2}}\left(e^{ix}-e^{-ix}\right)=i\sin x\\\cosh(x+iy)&=\cosh(x)\cos(y)+i\sinh(x)\sin(y)\\\sinh(x+iy)&=\sinh(x)\cos(y)+i\cosh(x)\sin(y)\\\tanh(ix)&=i\tan x\\\cosh x&=\cos(ix)\\\sinh x&=-i\sin(ix)\\\tanh x&=-i\tan(ix)\end{aligned}}}
chiu-kî hâm-sò͘ , chiu-kî sī
2
π
i
{\displaystyle 2\pi i}
π
i
{\displaystyle \pi i}
Ho̍k-pêⁿ-bīn téng ê Siang-khiok hâm-sò͘
sinh
(
z
)
{\displaystyle \sinh(z)}
cosh
(
z
)
{\displaystyle \cosh(z)}
tanh
(
z
)
{\displaystyle \tanh(z)}
coth
(
z
)
{\displaystyle \coth(z)}
sech
(
z
)
{\displaystyle \operatorname {sech} (z)}
csch
(
z
)
{\displaystyle \operatorname {csch} (z)}
↑ Weisstein, Eric W. "Hyperbolic Functions" . mathworld.wolfram.com (ēng Eng-gí). 2020-08-29 khòaⁿ--ê . ↑ (1999) Collins Concise Dictionary , 4th edition, HarperCollins, Glasgow, ISBN 0 00 472257 4 , p. 1386
↑ Collins Concise Dictionary , p. 328↑ "Hyperbolic Functions" . www.mathsisfun.com . 2020-08-29 khòaⁿ--ê . ↑ Collins Concise Dictionary , p. 1520↑ Ín-iōng chhò-gō͘: Bû-hāu ê <ref> tag;
chhōe bô chí-miâ ê ref bûn-jī Collins Concise Dictionary p. 3282 ↑ Collins Concise Dictionary , p. 1340↑ Collins Concise Dictionary , p. 329↑ tanh ↑ Ín-iōng chhò-gō͘: Bû-hāu ê <ref> tag;
chhōe bô chí-miâ ê ref bûn-jī :13 ↑ Ín-iōng chhò-gō͘: Bû-hāu ê <ref> tag;
chhōe bô chí-miâ ê ref bûn-jī :22 ↑ Ín-iōng chhò-gō͘: Bû-hāu ê <ref> tag;
chhōe bô chí-miâ ê ref bûn-jī :14 ↑ Ín-iōng chhò-gō͘: Bû-hāu ê <ref> tag;
chhōe bô chí-miâ ê ref bûn-jī :1 ↑ Martin, George E. (1986). The foundations of geometry and the non-euclidean plane (1st corr. pán.). New York: Springer-Verlag. p. 416. ISBN 3-540-90694-0 . ↑ "Prove the identity tanh(x/2) = (cosh(x) - 1)/sinh(x)" . StackExchange (mathematics). 24 January 2016 khòaⁿ--ê . ↑ Audibert, Jean-Yves (2009). "Fast learning rates in statistical inference through aggregation". The Annals of Statistics. p. 1627. [1]
![{\displaystyle {\begin{aligned}\sinh(x+y)&=\sinh x\cosh y+\cosh x\sinh y\\\cosh(x+y)&=\cosh x\cosh y+\sinh x\sinh y\\[6px]\tanh(x+y)&={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a343fcb86c5fa15cceb217f4e92410c70a6e200)
![{\displaystyle {\begin{aligned}\sinh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\sqrt {2(\cosh x+1)}}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{2}}}\\[6px]\cosh \left({\frac {x}{2}}\right)&={\sqrt {\frac {\cosh x+1}{2}}}\\[6px]\tanh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\cosh x+1}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{\cosh x+1}}}={\frac {e^{x}-1}{e^{x}+1}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/412a4ffd109486f684e515634b33447b13444954)
