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第一章　函数　极限　连续

y. O. x. 第一章　函数　极限　连续. 第六节　双曲函数. 双曲正弦函数. y = ch x. y = sh x. 双曲余弦函数. y. O. x. 双曲正切函数. y = th x. y. O. x. 双曲余切函数. y = coth x. 这些函数之间存在着下述关系：. sh ( x  y ). = sh x ch y  ch x sh y. = ch x ch y  sh x sh y. ch ( x  y ). sh 2 x. = 2sh x ch x.

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第一章　函数　极限　连续

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  1. y O x 第一章　函数　极限　连续 第六节　双曲函数 双曲正弦函数 y = ch x y = sh x 双曲余弦函数

  2. y O x 双曲正切函数 y = th x

  3. y O x 双曲余切函数 y = coth x

  4. 这些函数之间存在着下述关系： sh (x y) = sh x ch y  ch x sh y. = ch x ch y sh x sh y. ch (x y) sh 2x =2sh x ch x. ch 2x =ch2x + sh2x. ch2x - sh2x = 1.

  5. 由双曲函数定义可得 我们来证明第一个公式. sh xch y + ch xsh y

  6. 　　　　　　　　　　　　　　　所以函数 y = ch x是偶函数 ； 因为 所以函数 y= sh x，y= th x，y = coth x为奇函数.

  7. 双曲函数的反函数叫做反双曲函数，分别 记为 arsh x，arch x，arth x， arcoth x . 反双曲函数还有如下的表达式：

  8. 下面我们给出公式 y = arsh x的推导： 于是可得 解之得 即 因为 u = ex> 0，所以上式取正号， 故 y = sh x的反函数为

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