Hyperbolic functions and inverses




The hyperbolic functions

The hyperbolic functions are defined in terms of ex and e-x.
 
                 ex - e-x
      sinh(x) = ------------
                    2

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

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

      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) = ea+b  =  ea.eb  =  (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)

Graphs

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 = (ey - e-y)/2
<=>
      x = (e2y - 1)/(2.ey)
<=>
      e2y - 2x ey - 1 = 0
<=>
      ey is the positive root of previous quadratic equation
<=>
      ey = x + sqrt(x2+1)
<=>
      y = ln( x + sqrt(x2+1) )
From this we have

 
      argsinh(x) = ln( x + sqrt(x2 + 1) )

The argcosh function

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 = (ey + e-y)/2  with y > 0
<=>
      x = (e2y + 1)/(2.ey)  with y > 0
<=>
      e2y - 2x ey + 1 = 0  with y > 0
<=>
      ey = 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)
<=>
          ey - e-y
      x = ----------
          ey + e-y
<=>
          e2y - 1
      x = ------------
          e2y + 1
<=>
      ...
<=>
             1 + x
      e2y = ---------
             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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