Hyperbolic functions and inverses

Themes > Science > Mathematics > Algebra > Foci of a conic section > Topics and Problems > Hyperbolic functions and inverses

..The hyperbolic functions

..Properties

..Sum formulas

..Graphs

..Inverse hyperbolic functions

..The argsinh function
..The argcosh function
..The argtanh function

..Derivatives

The hyperbolic functions

The hyperbolic functions are defined in terms of e^x and e^-x.

 
                 e^x - e^-x
      sinh(x) = ------------
                    2

                 e^x + e^-x
      cosh(x) = ------------
                    2

                sinh(x)    e^x - e^-x
      tanh(x) = ------- =  ----------
                cosh(x)    e^x + e^-x


      coth(x) = 1/ tanh(x)


      csch(x) = 1/ sinh(x)


      sech(x) = 1/ cosh(x)

Properties

It is easy to prove that

 
      cosh(-x)  = cosh(x)

      sinh(-x) = - sinh(x)

      tanh(-x) = - tanh(x)

      cosh(x) > 0

      cosh(x) + sinh(x) = e^x

      cosh(x) - sinh(x) = e^-x > 0

      cosh²(x) - sinh²(x) = 1

      cosh(x) = + sqrt( 1 + sinh²(x) )  >=  1

      1 - tanh²(x) = 1/cosh²(x)

                       1
      cosh(x) = ------------------
                sqrt(1 - tanh²(x))


                   tanh(x)
       sinh(x) = -------------------
                 sqrt(1 - tanh²(x))

Sum formulas

From previous properties, we deduce

 

cosh(a + b) + sinh(a + b) = e^a+b  =  e^a.e^b  =  (ch(a) + sh(a))(ch(b) + sh(b))
cosh(a + b) - sinh(a + b) = e^-(a+b) = e^-a.e^-b = (ch(a) - sh(a))(ch(b) - sh(b))

Adding and subtracting, we find

cosh(a + b) = cosh(a)cosh(b) + sinh(a)sinh(b) sinh(a + b) = sinh(a)cosh(b) + cosh(a)sinh(b)

Analogous

cosh(a - b) = cosh(a)cosh(b) - sinh(a)sinh(b) sinh(a - b) = sinh(a)cosh(b) - cosh(a)sinh(b)

Dividing, we have

 
               sinh(a)cosh(b) + cosh(a)sinh(b)
tanh(a + b) = ---------------------------------
               cosh(a)cosh(b) + sinh(a)sinh(b)

             = ...

 
                tanh(a) + tanh(b)
tanh(a + b)  =  --------------------
                1 + tanh(a) tanh(b)

Analogous

tanh(a) - tanh(b) tanh(a - b) = -------------------- 1 - tanh(a) tanh(b)

When a = b we have

cosh(2a) = cosh²(a) + sinh²(a) sinh(2a) = 2 sinh(a)cosh(a) 2 tanh(a) tanh(2a) = ------------- 1 + tanh²(a)

Inverse hyperbolic functions

The argsinh function

The function argsinh(x) is the inverse function of the function sinh(x).

 
      y = argsinh(x)
<=>
      x = sinh(y)
<=>
      x = (e^y - e^-y)/2
<=>
      x = (e^2y - 1)/(2.e^y)
<=>
      e^2y - 2x e^y - 1 = 0
<=>
      e^y is the positive root of previous quadratic equation
<=>
      e^y = x + sqrt(x²+1)
<=>
      y = ln( x + sqrt(x²+1) )

From this we have

argsinh(x) = ln( x + sqrt(x² + 1) )

On previous graph, it is easy to see that the inverse relation of cosh(x) is not a function.
Therefore we restrict the domain of cosh(x) to positive x-values.
Now the inverse function exists and we call that function argcosh(x).

 
      y = argcosh(x)
<=>
      x = cosh(y)  with y > 0
<=>
      x = (e^y + e^-y)/2  with y > 0
<=>
      x = (e^2y + 1)/(2.e^y)  with y > 0
<=>
      e^2y - 2x e^y + 1 = 0  with y > 0
<=>
      e^y = x + sqrt(x²-1)
<=>
      y = ln( x + sqrt(x² - 1) )

From this we have

argcosh(x) = ln( x + sqrt(x² - 1) )

The argtanh function

The function argtanh(x) is the inverse function of the function tanh(x).

 
      y = argtanh(x)
<=>
          e^y - e^-y
      x = ----------
          e^y + e^-y
<=>
          e^2y - 1
      x = ------------
          e^2y + 1
<=>
      ...
<=>
             1 + x
      e^2y = ---------
             1 - x
<=>
                    1 + x
      y = (1/2) ln ------
                    1 - x

From this we have

1 + x argtanh(x) = (1/2) ln ------ 1 - x

Derivatives

Appealing on the definitions of the hyperbolic functions, the formulas for differentiation and the chain rule, it is easy to show that

d -- sinh(u) = cosh(u) . u' dx d -- cosh(u) = sinh(u) . u' dx d -- tanh(u) = u'/ cosh²(u) dx d -- coth(u) = - u'/ sinh²(u) dx d -- argsinh(u) = u'/sqrt(u² + 1) dx d -- argcosh(u) = u'/sqrt(u² - 1) dx d -- argtanh(u) = u'/(1-u²) dx

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