Siang-khiok hâm-sò͘

連結=//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Sinh_cosh_tanh.svg/300px-Sinh_cosh_tanh.svg.png|縮圖|300x300像素
Sò͘-ha̍k siōng, **siang-khiok hâm-sò͘** () 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  chia-ê tiám hêng-sêng chi̍t ê tan-ūi-îⁿ kāng-khoán,  chia-ê tiám ē tan-ūi siang-khiok-sòaⁿ ê chiàⁿ-chhiú pòaⁿ-pêng. Koh ū chi̍t ê sio-siâng ê só͘-chāi sī,  kap  ê tō-hâm-sò͘ hun-pia̍t sī  kap , ah  kap  ê tō-hâm-só͘ hun-pia̍t sī   kap .

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** "" (),
* **siang-khiok cosine** "" (),

tùi chit nn̄g ê ē-sái tit-tio̍h:

* **siang-khiok** **tangent** "" (),
* **siang-khiok** **cosecant** "" or "" ()
* **siang-khiok** **secant** "" (),
* **siang-khiok** **cotangent** "" ().

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

### Chí-sò͘ hâm-sò͘
[連結=//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Hyperbolic_and_exponential%3B_sinh.svg/220px-Hyperbolic_and_exponential%3B_sinh.svg.png|右|縮圖| sī  kap  [Kiám-hoat|chha ê chi̍t-pòaⁿ]]
[連結=//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Hyperbolic_and_exponential%3B_cosh.svg/220px-Hyperbolic_and_exponential%3B_cosh.svg.png|右|縮圖| sī  kap  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 = \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 = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}.
* Siang-khiok tangent: \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:  ê sî, \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:  \operatorname{sech} x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}}
= \frac{2e^x} {e^{2x} + 1}.
* Siang-khiok cosecant:  ê sî,  \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).
* Siang-khiok cosine: \cosh x = \cos (i x).
* Siang-khiok tangent: \tanh x = -i \tan (i x).
* Siang-khiok cotangent: \coth x = i \cot (i x).
* Siang-khiok secant:  \operatorname{sech} x = \sec (i x).
* Siang-khiok cosecant:\operatorname{csch} x = i \csc (i x).

kî-tiong  sī hi-sò͘ tan-ūi-goân, .

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
 kap  hun-pia̍t sī khia-sò͘ hâm-sò͘ kap siang-sò͘ hâm-sò͘:\begin{align}
 \sinh (-x) &= -\sinh x \\
 \cosh (-x) &=  \cosh x
\end{align}Só͘-í:\begin{align}
               \tanh (-x) &= -\tanh x \\
               \coth (-x) &= -\coth x \\
 \operatorname{sech} (-x) &=  \operatorname{sech} x \\
 \operatorname{csch} (-x) &= -\operatorname{csch} x
\end{align}Só͘-í,  kap  sī siang-sò͘ hâm-sò͘; kî-thaⁿ-·ê sī khia-sò͘ hâm-sò͘.\begin{align}
 \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{align}\begin{align}
 \cosh x + \sinh x &= e^x \\
 \cosh x - \sinh x &= e^{-x} \\
 \cosh^2 x - \sinh^2 x &= 1
\end{align}siōng-bóe chi̍t ê kap Pythagorean saⁿ-kak hêng-têng-sek ū sêng.

Lán mā ū\begin{align}
 \operatorname{sech} ^{2} x &= 1 - \tanh^{2} x \\
 \operatorname{csch} ^{2} x &= \coth^{2} x - 1
\end{align}

### Piàn-sò͘ saⁿ-thiⁿ
\begin{align}
 \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{align}te̍k-pia̍t sī\begin{align}
\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{align}Koh ū:\begin{align}
 \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{align}

### Piàn-sò͘ saⁿ-kiám
\begin{align}
 \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{align}Koh ū:\begin{align}
 \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{align}

### Piàn-sò͘ kiám-pòaⁿ
\begin{align}
 \sinh\left(\frac{x}{2}\right) &= \frac{\sinh x}{\sqrt{2 (\cosh x + 1)} } &&= \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} &&= \sgn x \, \sqrt \frac{\cosh x-1}{\cosh x+1} = \frac{e^x - 1}{e^x + 1}
\end{align}kî-tiong  sī chèng-hū-hō hâm-sò͘.

 ê sî, \tanh\left(\frac{x}{2}\right) = \frac{\cosh x - 1}{\sinh x} = \coth x - \operatorname{csch} x 

### Pêng-hong kong-sek
\begin{align}
\sinh^2 x &= \frac{1}{2}(\cosh 2x -1) \\
\cosh^2 x &= \frac{1}{2}(\cosh 2x +1)
\end{align}

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

## Ēng tùi-sò͘ piat-ta̍t hoán-siang-khiok hâm-sò͘
 \begin{align}
 \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 {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 

## Tō-hâm-sò͘
\begin{align}
 \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{align}\begin{align}
 \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 

## Jī-chhù tō-hâm-sò͘
 kap  ê jī-chhù tō-hâm-sò͘ tiō sī in ka-kī: \frac{d^2}{dx^2}\sinh x = \sinh x  \frac{d^2}{dx^2}\cosh x = \cosh x \, .Só͘-ū ū chit khoán sèng-chit ê hâm-sò͘ lóng sī  kap  ê sòaⁿ-sèng cho͘-ha̍p, te̍k-pia̍t pau-koat chí-sò͘ hâm-sò͘  e^x  kap  e^{-x} .

## Phiau-chún chek-hun
\begin{align}
 \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{align}Chia-ê chek-hun-sek lóng ē-sái ēng siang-khiok tāi-thè-hoat lâi chèng-bêng:\begin{align}
 \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} &= \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}{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{align}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-seke^x = \cosh x + \sinh x,kape^{-x} = \cosh x - \sinh x.Kap Euler kong-sek ha̍p-pènge^{ix} = \cos x + i\sin x,tiō ē tit-tio̍he^{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 ūe^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 \begin{align}
  e^{i x} &= \cos x + i \sin x \\
 e^{-i x} &= \cos x - i \sin x
\end{align}só͘-í:\begin{align}
   \cosh(ix) &= \frac{1}{2} \left(e^{i x} + e^{-i x}\right) = \cos x \\
   \sinh(ix) &= \frac{1}{2} \left(e^{i x} - e^{-i x}\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{align}Só͘-í, siang-khiok hâm-sò͘ tùi in ê hi-pō͘ lâi kóng, sī chiu-kî hâm-sò͘, chiu-kî sī 2 \pi i (iah-m̄-koh siang-khiok tangent kap cotangent ê chiu-kî sī \pi i).

## Chham-khó chu-liāu

Lūi-pia̍t:Siang-khiok hâm-sò͘