Understanding Hyperbolic Functions and Their Inverses
This content delves into the world of hyperbolic functions, discussing their formation from exponential functions, identities, derivatives, and inverse hyperbolic functions. The text explores crucial concepts such as hyperbolic trigonometric identities, derivatives of hyperbolic functions, and integrals involving inverse hyperbolic functions.
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Hyperbolic Functions The hyperbolic functions are formed by taking combinations of the two exponential functions ??,??? ? ?
Identities for hyperbolic functions Derivatives of hyperbolic functions
? ?? ???????:???? sinh? ???? ??+ ? ??? 2 ?? ? ? 2 ??+ ? ? 2 ? ?? ? ?? =?? ?? ?? sinh? = = = cosh??? ?? ? ?? ???????: ???? csch? cosh??? ? ?? 1 = coth?csch??? ?? = = ??? 2? sinh? ?? 1 ??? 2? ?? ???????:???? 0 1 1cosh2? 1 1 ?? =1 cosh2? 1 ?? =1 sinh2? 2 = 2 ? 2 2 0 0 0 =sinh2 1 2 0.40672 4
ln 2 4??sinh? ?? ???????: 0 ln 2 ln 2 4???? ? ? 2 ?2? ?0dx = ?? = 2 0 0 ln 2 ln 2 2?2? 2 ?? = ?2? 2?0 = 0 = ?2??2 2??2 1 1.6137 ???????:???????? tanh? + tanh? ??? 2? ?? = ??? ? ??? ??? + tanh? ??? 2??? ?????? ? = cosh? ?? = sinh??? ?????? ? = tanh? ?? = ??? 2? ?? = ?? ?+ ?1/2?? = ?? ??? ? +(tanh?)3/2 + ? 3/2
Inverse Hyperbolic functions The inverse of the six hyperbolic functions are very useful in integration. We denote its inverse by ? = ??? 1? < ? < The restricted function ? = ??? 1? , 1 ??? ? ?? ??????? .??????? ?? ? = ??? 1? ? 1 ??? ? ? ???????? ? = sech? = The hyperbolic tangent, cotangent and cosecant are have inverses ? = ??? 1?,? = ??? 1? ,? = ??? 1?
Identities for inverse hyperbolic functions To prove that ??? 1? = ??? 11 ? ??? ? = ??? 11 cosh? =1 ? ? = ??? 1? ??? 1? = ??? 11 ? ? = sech? ?
Derivatives of inverse hyperbolic functions Integral to inverse Hyperbolic functions 1 2?? ???????:???????? 3 + 4?2 0 ?????? ? = 3 ,? = 2? 1 1 ?? 2? 2 ?? = 2?? ?2+ ?2= ??? 1 = ??? 1 ??? 10 = 3 3 0 0 0.98665
???????: ????? ??? ???1? = ln ? + ?2+ 1 ??? ? = ??? 1? sinh? = ? ?? ? ? 2 = ? 1 ??= 2? ?? ? ?= 2? ?? ?2? 1 = 2??? ?2? 2??? 1 = 0 ??=2? 4?2+ 4 2 ??> 0 ??????? ????? ???? ??= ? + ?2+ 1 = ? ?2+ 1 ?2+ 1 ? = ln ? + ??? 1? = ln ? + ?2+ 1 sin? 1 + ???2??? ???????: ? = cos? ?? = sin??? ?? 1 + ?2= ??? 1? + ? = ??? 1(cos?) + ? =
????1? ln? ???????: ???? lim ? = cosh? =??+ ? ? ??+ ? ?= 2? ?2?+ 1 = 2??? ?2? 2???+ 1 = 0 ??=2? 4?2 4 2 ??= ? + ? = ??? 1? 2 ?2 1 = ? ?2 1 ?2 1 ? = ln ? + ??? 1? = ?? ? + ?2 1 ?2 1 ln? = lim ? ln ? + ln? + ?2 1 = lim ? ? 1 1 1 + ?2 = lim ? ln = ln2 1