Sò͘-ha̍k siōng, **siang-khiok hâm-sò͘** (Hàn-jī: 雙曲函數) sī 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:

* **siang-khiok sine** "sinh" (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/),
* **siang-khiok cosine** "cosh" (/ˈkɒʃ, ˈkoʊʃ/),

tùi chit nn̄g ê ē-sái tit-tio̍h:

* **siang-khiok** **tangent** "tanh" (/ˈtæŋ, ˈtæntʃ, ˈθæn/),
* **siang-khiok** **cosecant** "csch" or "cosech" (/ˈkoʊsɛtʃ, ˈkoʊʃɛk/)
* **siang-khiok** **secant** "sech" (/ˈsɛtʃ, ˈʃɛk/),
* **siang-khiok** **cotangent** "coth" (/ˈkɒθ, ˈkoʊθ/).

Tēng-gī
-------

Beh tēng-gī siang-khiok hâm-sò͘, ū chiok chē chióng hoat.

### Chí-sò͘ hâm-sò͘

sinh *x* sī *ex* kap *e*−*x* chha ê chi̍t-pòaⁿ
cosh *x* sī *ex* kap *e*−*x* ê pêng-kun
Tùi chí-sò͘ hâm-sò͘ lâi tēng-gī:

* 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}}}.}

{\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}}}.}

{\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}}.}

{\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 ê sî, 

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}}.}

{\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}}.}

{\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 ê sî, 

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}}.}

{\displaystyle \operatorname {csch} x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}.}

### Ho̍k-cha̍p-sò͘ saⁿ-kak hâm-sò͘

Tùi ho̍k-cha̍p-sò͘ piān-só͘ ê saⁿ-kak hâm-sò͘ mā ē-sái lâi tēng-gī siang-khiok hâm-sò͘:

* Siang-khiok sine: 

sinh
⁡
x
=
−
i
sin
⁡
(
i
x
)
.

{\displaystyle \sinh x=-i\sin(ix).}

{\displaystyle \sinh x=-i\sin(ix).}
* Siang-khiok cosine: 

cosh
⁡
x
=
cos
⁡
(
i
x
)
.

{\displaystyle \cosh x=\cos(ix).}

{\displaystyle \cosh x=\cos(ix).}
* Siang-khiok tangent: 

tanh
⁡
x
=
−
i
tan
⁡
(
i
x
)
.

{\displaystyle \tanh x=-i\tan(ix).}

{\displaystyle \tanh x=-i\tan(ix).}
* Siang-khiok cotangent: 

coth
⁡
x
=
i
cot
⁡
(
i
x
)
.

{\displaystyle \coth x=i\cot(ix).}

{\displaystyle \coth x=i\cot(ix).}
* Siang-khiok secant: 

sech
⁡
x
=
sec
⁡
(
i
x
)
.

{\displaystyle \operatorname {sech} x=\sec(ix).}

{\displaystyle \operatorname {sech} x=\sec(ix).}
* Siang-khiok cosecant:

csch
⁡
x
=
i
csc
⁡
(
i
x
)
.

{\displaystyle \operatorname {csch} x=i\csc(ix).}

{\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.

Ū lō͘-ēng ê koan-hē-sek
-----------------------

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}}}

{\displaystyle {\begin{aligned}\sinh(-x)&=-\sinh x\\\cosh(-x)&=\cosh x\end{aligned}}}Só͘-í:



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}}}

{\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}}}Só͘-í, 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}}}

{\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}}}

{\displaystyle {\begin{aligned}\cosh x+\sinh x&=e^{x}\\\cosh x-\sinh x&=e^{-x}\\\cosh ^{2}x-\sinh ^{2}x&=1\end{aligned}}}siōng-bóe chi̍t ê kap 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}}}

{\displaystyle {\begin{aligned}\operatorname {sech} ^{2}x&=1-\tanh ^{2}x\\\operatorname {csch} ^{2}x&=\coth ^{2}x-1\end{aligned}}}
### Piàn-sò͘ saⁿ-thiⁿ




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}}}

{\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}}}te̍k-pia̍t sī



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}}}

{\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}}}Koh ū:



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}}}

{\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}}}
### Piàn-sò͘ saⁿ-kiám




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}}}

{\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}}}Koh ū:



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}}}

{\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}}}
### Piàn-sò͘ kiám-pòaⁿ




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}}}

{\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}}}kî-tiong sgn sī chèng-hū-hō hâm-sò͘.
*x* ≠ 0 ê sî,


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}

{\displaystyle \tanh \left({\frac {x}{2}}\right)={\frac {\cosh x-1}{\sinh x}}=\coth x-\operatorname {csch} x}
### Pêng-hong kong-sek



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}}}

{\displaystyle {\begin{aligned}\sinh ^{2}x&={\frac {1}{2}}(\cosh 2x-1)\\\cosh ^{2}x&={\frac {1}{2}}(\cosh 2x+1)\end{aligned}}}
### Put-téng-sek

Chit ê put-téng-sek tī thóng-kè-ha̍k lāi-té chin ū lō͘-ēng:
{\displaystyle \operatorname {cosh} (t)\leq e^{t^{2}/2}}.

Ēng tùi-sò͘ piat-ta̍t hoán-siang-khiok hâm-sò͘
----------------------------------------------

 



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}}}

{\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}}}

Tō-hâm-sò͘
----------




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}}}

{\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}}}

{\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}}}

Jī-chhù tō-hâm-sò͘
------------------

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}

{\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\,.}

{\displaystyle {\frac {d^{2}}{dx^{2}}}\cosh x=\cosh x\,.}Só͘-ū ū chit khoán sèng-chit ê hâm-sò͘ lóng sī sinh kap cosh ê sòaⁿ-sèng cho͘-ha̍p, te̍k-pia̍t pau-koat chí-sò͘ hâm-sò͘ {\displaystyle e^{x}} kap {\displaystyle e^{-x}}.

Phiau-chún chek-hun
-------------------




∫
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}}}

{\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}}}Chia-ê chek-hun-sek lóng ē-sái ēng siang-khiok tāi-thè-hoat lâi chèng-bêng:



∫

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}}}

{\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}}}kî-tiong *C* sī chek-hun siông-sò͘.

Kap chí-sò͘ hâm-sò͘ ê koan-hē
-----------------------------

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,}

{\displaystyle e^{x}=\cosh x+\sinh x,}kap



e

−
x

=
cosh
⁡
x
−
sinh
⁡
x
.

{\displaystyle e^{-x}=\cosh x-\sinh x.}

{\displaystyle e^{-x}=\cosh x-\sinh x.}Kap Euler kong-sek ha̍p-pèng



e

i
x

=
cos
⁡
x
+
i
sin
⁡
x
,

{\displaystyle e^{ix}=\cos x+i\sin x,}

{\displaystyle e^{ix}=\cos x+i\sin x,}tiō ē tit-tio̍h



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)}

{\displaystyle e^{x+iy}=(\cosh x+\sinh x)(\cos y+i\sin y)}che tiō sī 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}}}}}

{\displaystyle e^{x}={\sqrt {\frac {1+\tanh x}{1-\tanh x}}}={\frac {1+\tanh {\frac {x}{2}}}{1-\tanh {\frac {x}{2}}}}}

Ho̍k-cha̍p-sò͘ piān-só͘ ê siang-khiok hâm-sò͘
---------------------------------------------

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}}}

{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\e^{-ix}&=\cos x-i\sin x\end{aligned}}}só͘-í:



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}}}

{\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}}}Só͘-í, siang-khiok hâm-sò͘ tùi in ê hi-pō͘ lâi kóng, sī chiu-kî hâm-sò͘, chiu-kî sī {\displaystyle 2\pi i} (iah-m̄-koh siang-khiok tangent kap cotangent ê chiu-kî sī {\displaystyle \pi i}).

Ho̍k-pêⁿ-bīn téng ê Siang-khiok hâm-sò͘|  |  |  |  |  |  |
| {\displaystyle \sinh(z)} | {\displaystyle \cosh(z)} | {\displaystyle \tanh(z)} | {\displaystyle \coth(z)} | {\displaystyle \operatorname {sech} (z)} | {\displaystyle \operatorname {csch} (z)} |

Chham-khó chu-liāu
------------------

1. ↑  Weisstein, Eric W. "Hyperbolic Functions". *mathworld.wolfram.com* (ēng Eng-gí). 2020-08-29 khòaⁿ--ê.
2. ↑  (1999) *Collins Concise Dictionary*, 4th edition, HarperCollins, Glasgow, ISBN 0 00 472257 4, p. 1386
3. ↑  *Collins Concise Dictionary*, p. 328
4. ↑  "Hyperbolic Functions". *www.mathsisfun.com*. 2020-08-29 khòaⁿ--ê.
5. ↑  *Collins Concise Dictionary*, p. 1520
6. ↑
7. ↑  *Collins Concise Dictionary*, p. 1340
8. ↑  *Collins Concise Dictionary*, p. 329
9. ↑  tanh
10. ↑
11. ↑
12. ↑
13. ↑
14. ↑  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.
15. ↑  "Prove the identity tanh(x/2) = (cosh(x) - 1)/sinh(x)". *StackExchange (mathematics)*. 24 January 2016 khòaⁿ--ê.
16. ↑  Audibert, Jean-Yves (2009). "Fast learning rates in statistical inference through aggregation". The Annals of Statistics. p. 1627.