Siang-khiok hâm-sò͘

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 e^x kap e^−x chha ê chi̍t-pòaⁿ

cosh x sī e^x 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,
  $$\sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.$$
• 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: x ≠ 0 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: x ≠ 0 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:
  $$\sinh x=-i\sin(ix).$$
• Siang-khiok cosine:
  $$\cosh x=\cos(ix).$$
• Siang-khiok tangent:
  $$\tanh x=-i\tan(ix).$$
• Siang-khiok cotangent:
  $$\coth x=i\cot(ix).$$
• Siang-khiok secant:
  $$\operatorname {sech} x=\sec(ix).$$
• Siang-khiok cosecant:
  $$\operatorname {csch} x=i\csc(ix).$$

kî-tiong i sī hi-sò͘ tan-ūi-goân, i² = −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:

$${\begin{aligned}\sinh(-x)&=-\sinh x\\\cosh(-x)&=\cosh x\end{aligned}}$$

Só͘-í:

$${\begin{aligned}\tanh(-x)&=-\tanh x\\\coth(-x)&=-\coth x\\\operatorname {sech} (-x)&=\operatorname {sech} x\\\operatorname {csch} (-x)&=-\operatorname {csch} x\end{aligned}}$$

Só͘-í, cosh x kap sech x sī siang-sò͘ hâm-sò͘; kî-thaⁿ-·ê sī khia-sò͘ hâm-sò͘.

$${\begin{aligned}\operatorname {arsech} x&=\operatorname {arcosh} \left({\frac {1}{x}}\right)\\\operatorname {arcsch} x&=\operatorname {arsinh} \left({\frac {1}{x}}\right)\\\operatorname {arcoth} x&=\operatorname {artanh} \left({\frac {1}{x}}\right)\end{aligned}}$$

$${\begin{aligned}\cosh x+\sinh x&=e^{x}\\\cosh x-\sinh x&=e^{-x}\\\cosh ^{2}x-\sinh ^{2}x&=1\end{aligned}}$$

siōng-bóe chi̍t ê kap Pythagorean saⁿ-kak hêng-têng-sek ū sêng.

Lán mā ū

$${\begin{aligned}\operatorname {sech} ^{2}x&=1-\tanh ^{2}x\\\operatorname {csch} ^{2}x&=\coth ^{2}x-1\end{aligned}}$$

Piàn-sò͘ saⁿ-thiⁿ

$${\begin{aligned}\sinh(x+y)&=\sinh x\cosh y+\cosh x\sinh y\\\cosh(x+y)&=\cosh x\cosh y+\sinh x\sinh y\\[6px]\tanh(x+y)&={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\\\end{aligned}}$$

te̍k-pia̍t sī

$${\begin{aligned}\cosh(2x)&=\sinh ^{2}{x}+\cosh ^{2}{x}=2\sinh ^{2}x+1=2\cosh ^{2}x-1\\\sinh(2x)&=2\sinh x\cosh x\\\tanh(2x)&={\frac {2\tanh x}{1+\tanh ^{2}x}}\\\end{aligned}}$$

Koh ū:

$${\begin{aligned}\sinh x+\sinh y&=2\sinh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\cosh x+\cosh y&=2\cosh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\end{aligned}}$$

Piàn-sò͘ saⁿ-kiám

$${\begin{aligned}\sinh(x-y)&=\sinh x\cosh y-\cosh x\sinh y\\\cosh(x-y)&=\cosh x\cosh y-\sinh x\sinh y\\\tanh(x-y)&={\frac {\tanh x-\tanh y}{1-\tanh x\tanh y}}\\\end{aligned}}$$

Koh ū:

$${\begin{aligned}\sinh x-\sinh y&=2\cosh \left({\frac {x+y}{2}}\right)\sinh \left({\frac {x-y}{2}}\right)\\\cosh x-\cosh y&=2\sinh \left({\frac {x+y}{2}}\right)\sinh \left({\frac {x-y}{2}}\right)\\\end{aligned}}$$

Piàn-sò͘ kiám-pòaⁿ

$${\begin{aligned}\sinh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\sqrt {2(\cosh x+1)}}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{2}}}\\[6px]\cosh \left({\frac {x}{2}}\right)&={\sqrt {\frac {\cosh x+1}{2}}}\\[6px]\tanh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\cosh x+1}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{\cosh x+1}}}={\frac {e^{x}-1}{e^{x}+1}}\end{aligned}}$$

kî-tiong sgn sī chèng-hū-hō hâm-sò͘.

x ≠ 0 ê sî,

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

Pêng-hong kong-sek

$${\begin{aligned}\sinh ^{2}x&={\frac {1}{2}}(\cosh 2x-1)\\\cosh ^{2}x&={\frac {1}{2}}(\cosh 2x+1)\end{aligned}}$$

Put-téng-sek

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

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

 

$${\begin{aligned}\operatorname {arsinh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right)&&x\geq 1\\\operatorname {artanh} (x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)&&|x|<1\\\operatorname {arcoth} (x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)&&|x|>1\\\operatorname {arsech} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}-1}}\right)=\ln \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)&&0<x\leq 1\\\operatorname {arcsch} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}+1}}\right)&&x\neq 0\end{aligned}}$$

Tō-hâm-sò͘

$${\begin{aligned}{\frac {d}{dx}}\sinh x&=\cosh x\\{\frac {d}{dx}}\cosh x&=\sinh x\\{\frac {d}{dx}}\tanh x&=1-\tanh ^{2}x=\operatorname {sech} ^{2}x={\frac {1}{\cosh ^{2}x}}\\{\frac {d}{dx}}\coth x&=1-\coth ^{2}x=-\operatorname {csch} ^{2}x=-{\frac {1}{\sinh ^{2}x}}&&x\neq 0\\{\frac {d}{dx}}\operatorname {sech} x&=-\tanh x\operatorname {sech} x\\{\frac {d}{dx}}\operatorname {csch} x&=-\coth x\operatorname {csch} x&&x\neq 0\end{aligned}}$$

$${\begin{aligned}{\frac {d}{dx}}\operatorname {arsinh} x&={\frac {1}{\sqrt {x^{2}+1}}}\\{\frac {d}{dx}}\operatorname {arcosh} x&={\frac {1}{\sqrt {x^{2}-1}}}&&1<x\\{\frac {d}{dx}}\operatorname {artanh} x&={\frac {1}{1-x^{2}}}&&|x|<1\\{\frac {d}{dx}}\operatorname {arcoth} x&={\frac {1}{1-x^{2}}}&&1<|x|\\{\frac {d}{dx}}\operatorname {arsech} x&=-{\frac {1}{x{\sqrt {1-x^{2}}}}}&&0<x<1\\{\frac {d}{dx}}\operatorname {arcsch} x&=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}&&x\neq 0\end{aligned}}$$

Jī-chhù tō-hâm-sò͘

sinh kap cosh ê jī-chhù tō-hâm-sò͘ tiō sī in ka-kī:

$${\frac {d^{2}}{dx^{2}}}\sinh x=\sinh x$$

$${\frac {d^{2}}{dx^{2}}}\cosh x=\cosh x\,.$$

Só͘-ū ū chit khoán sèng-chit ê hâm-sò͘ lóng sī sinh kap cosh ê sòaⁿ-sèng cho͘-ha̍p, te̍k-pia̍t pau-koat chí-sò͘ hâm-sò͘ $e^{x}$ kap $e^{-x}$.

Phiau-chún chek-hun

$${\begin{aligned}\int \sinh(ax)\,dx&=a^{-1}\cosh(ax)+C\\\int \cosh(ax)\,dx&=a^{-1}\sinh(ax)+C\\\int \tanh(ax)\,dx&=a^{-1}\ln(\cosh(ax))+C\\\int \coth(ax)\,dx&=a^{-1}\ln \left|\sinh(ax)\right|+C\\\int \operatorname {sech} (ax)\,dx&=a^{-1}\arctan(\sinh(ax))+C\\\int \operatorname {csch} (ax)\,dx&=a^{-1}\ln \left|\tanh \left({\frac {ax}{2}}\right)\right|+C=a^{-1}\ln \left|\coth \left(ax\right)-\operatorname {csch} \left(ax\right)\right|+C=-a^{-1}\operatorname {arcoth} \left(\cosh \left(ax\right)\right)+C\end{aligned}}$$

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

$${\begin{aligned}\int {{\frac {1}{\sqrt {a^{2}+u^{2}}}}\,du}&=\operatorname {arsinh} \left({\frac {u}{a}}\right)+C\\\int {{\frac {1}{\sqrt {u^{2}-a^{2}}}}\,du}&=\operatorname {sgn} {u}\operatorname {arcosh} \left|{\frac {u}{a}}\right|+C\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {artanh} \left({\frac {u}{a}}\right)+C&&u^{2}<a^{2}\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {arcoth} \left({\frac {u}{a}}\right)+C&&u^{2}>a^{2}\\\int {{\frac {1}{u{\sqrt {a^{2}-u^{2}}}}}\,du}&=-a^{-1}\operatorname {arsech} \left|{\frac {u}{a}}\right|+C\\\int {{\frac {1}{u{\sqrt {a^{2}+u^{2}}}}}\,du}&=-a^{-1}\operatorname {arcsch} \left|{\frac {u}{a}}\right|+C\end{aligned}}$$

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

$$e^{x}=\cosh x+\sinh x,$$

kap

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

Kap Euler kong-sek ha̍p-pèng

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

tiō ē tit-tio̍h

$$e^{x+iy}=(\cosh x+\sinh x)(\cos y+i\sin y)$$

che tiō sī it-poaⁿ ê ho̍k-cha̍p-sò͘ chí-sò͘ hâm-sò͘ ê kong-sek.

Lēng-gōa koh ū

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

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

$${\begin{aligned}e^{ix}&=\cos x+i\sin x\\e^{-ix}&=\cos x-i\sin x\end{aligned}}$$

só͘-í:

$${\begin{aligned}\cosh(ix)&={\frac {1}{2}}\left(e^{ix}+e^{-ix}\right)=\cos x\\\sinh(ix)&={\frac {1}{2}}\left(e^{ix}-e^{-ix}\right)=i\sin x\\\cosh(x+iy)&=\cosh(x)\cos(y)+i\sinh(x)\sin(y)\\\sinh(x+iy)&=\sinh(x)\cos(y)+i\cosh(x)\sin(y)\\\tanh(ix)&=i\tan x\\\cosh x&=\cos(ix)\\\sinh x&=-i\sin(ix)\\\tanh x&=-i\tan(ix)\end{aligned}}$$

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

Ho̍k-pêⁿ-bīn téng ê Siang-khiok hâm-sò͘

$\sinh(z)$	$\cosh(z)$	$\tanh(z)$	$\coth(z)$	$\operatorname {sech} (z)$	$\operatorname {csch} (z)$