Siang-khiok hâm-sò͘

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Sinh cosh tanh.svg

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-sekbu̍t-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]

tùi chit nn̄g ê ē-sái tit-tio̍h:[4]

Tēng-gī[siu-kái | kái goân-sí-bé]

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

Chí-sò͘ hâm-sò͘[siu-kái | kái goân-sí-bé]

sinh xex kap ex chha ê chi̍t-pòaⁿ
cosh xex kap ex ê pêng-kun

Tùi chí-sò͘ hâm-sò͘ lâi tēng-gī:[10][11]

  • Siang-khiok sine: chí-sò͘ hâm-sò͘ ê khia-pō͘, tiō-sī,
  • Siang-khiok cosine: the even part of the exponential function, that is,
  • Siang-khiok tangent:
  • Siang-khiok cotangent: x ≠ 0 ê sî,
  • Siang-khiok secant:
  • Siang-khiok cosecant: x ≠ 0 ê sî,

Ho̍k-cha̍p-sò͘ saⁿ-kak hâm-sò͘[siu-kái | kái goân-sí-bé]

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]
  • Siang-khiok cosine:[13]
  • Siang-khiok tangent:
  • Siang-khiok cotangent:
  • Siang-khiok secant:
  • Siang-khiok cosecant:

kî-tiong ihi-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[siu-kái | kái goân-sí-bé]

sinh kap cosh hun-pia̍t sī khia-sò͘ hâm-sò͘ kap siang-sò͘ hâm-sò͘:

Só͘-í:
Só͘-í, cosh x kap sech xsiang-sò͘ hâm-sò͘; kî-thaⁿ-·ê sī khia-sò͘ hâm-sò͘.
siōng-bóe chi̍t ê kap Pythagorean saⁿ-kak hêng-têng-sek ū sêng.

Lán mā ū

Piàn-sò͘ saⁿ-thiⁿ[siu-kái | kái goân-sí-bé]

te̍k-pia̍t sī
Koh ū:

Piàn-sò͘ saⁿ-kiám[siu-kái | kái goân-sí-bé]

Koh ū:[14]

Piàn-sò͘ kiám-pòaⁿ[siu-kái | kái goân-sí-bé]

kî-tiong sgnchèng-hū-hō hâm-sò͘.[15]

x ≠ 0 ê sî,

Pêng-hong kong-sek[siu-kái | kái goân-sí-bé]

Put-téng-sek[siu-kái | kái goân-sí-bé]

Chit ê put-téng-sekthóng-kè-ha̍k lāi-té chin ū lō͘-ēng: .[16]

Ēng tùi-sò͘ piat-ta̍t hoán-siang-khiok hâm-sò͘[siu-kái | kái goân-sí-bé]

 

Tō-hâm-sò͘[siu-kái | kái goân-sí-bé]

Jī-chhù tō-hâm-sò͘[siu-kái | kái goân-sí-bé]

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

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 kap .

Phiau-chún chek-hun[siu-kái | kái goân-sí-bé]

Chia-ê chek-hun-sek lóng ē-sái ēng siang-khiok tāi-thè-hoat lâi chèng-bêng:
kî-tiong Cchek-hun siông-sò͘.

Kap chí-sò͘ hâm-sò͘ ê koan-hē[siu-kái | kái goân-sí-bé]

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

kap
Kap Euler kong-sek ha̍p-pèng
tiō ē tit-tio̍h
che tiō sī it-poaⁿ ê ho̍k-cha̍p-sò͘ chí-sò͘ hâm-sò͘ ê kong-sek.

Lēng-gōa koh ū

Ho̍k-cha̍p-sò͘ piān-só͘ ê siang-khiok hâm-sò͘[siu-kái | kái goân-sí-bé]

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

só͘-í:
Só͘-í, siang-khiok hâm-sò͘ tùi in ê hi-pō͘ lâi kóng, sī chiu-kî hâm-sò͘, chiu-kî sī (iah-m̄-koh siang-khiok tangent kap cotangent ê chiu-kî sī ).

Ho̍k-pêⁿ-bīn téng ê Siang-khiok hâm-sò͘
Complex Sinh.jpg
Complex Cosh.jpg
Complex Tanh.jpg
Complex Coth.jpg
Complex Sech.jpg
Complex Csch.jpg

Chham-khó chu-liāu[siu-kái | kái goân-sí-bé]

  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. Ín-iōng chhò-gō͘: Bû-hāu ê <ref> tag; chhōe bô chí-miâ ê ref bûn-jī Collins Concise Dictionary p. 3282
  7. Collins Concise Dictionary, p. 1340
  8. Collins Concise Dictionary, p. 329
  9. tanh
  10. Ín-iōng chhò-gō͘: Bû-hāu ê <ref> tag; chhōe bô chí-miâ ê ref bûn-jī :13
  11. Ín-iōng chhò-gō͘: Bû-hāu ê <ref> tag; chhōe bô chí-miâ ê ref bûn-jī :22
  12. Ín-iōng chhò-gō͘: Bû-hāu ê <ref> tag; chhōe bô chí-miâ ê ref bûn-jī :14
  13. Ín-iōng chhò-gō͘: Bû-hāu ê <ref> tag; chhōe bô chí-miâ ê ref bûn-jī :1
  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.  [1]