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.

• Hyperbolic Functions
• Exponential Functions
• Trigonometric Identities
• Inverse Functions
• Derivatives

balfour Follow

Uploaded on Jul 24, 2024 | 0 Views

Download Presentation

Please find below an Image/Link to download the presentation.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.

Presentation Transcript

1. Hyperbolic Functions The hyperbolic functions are formed by taking combinations of the two exponential functions ??,??? ? ?

2. Identities for hyperbolic functions Derivatives of hyperbolic functions

3. ? ?? ???????:???? 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

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

5. 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?

6. Identities for inverse hyperbolic functions To prove that ??? 1? = ??? 11 ? ??? ? = ??? 11 cosh? =1 ? ? = ??? 1? ??? 1? = ??? 11 ? ? = sech? ?

7. 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

8. ???????: ????? ??? ???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?) + ? =

9. ????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