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arcsinh, arccosh, arctanh, arccsch, arcsech, arccoth -- the inverse hyperbolic functions

Introduction

arcsinh(x) represents the inverse of the sine function.

arccosh(x) represents the inverse of the cosine function.

arctanh(x) represents the inverse of the tangent function.

arccsch(x) represents the inverse of the cosecant function.

arcsech(x) represents the inverse of the secant function.

arccoth(x) represents the inverse of the cotangent function.

Call(s)

arcsinh(x)
arccosh(x)
arctanh(x)
arccsch(x)
arcsech(x)
arccoth(x)

Parameters

x - an arithmetical expression

Returns

an arithmetical expression.

Overloadable:

x

Side Effects

When called with a floating point argument, the functions are sensitive to the environment variable DIGITS which determines the numerical working precision.

Related Functions

sinh, cosh, tanh, csch, sech, coth

Details

Example 1

We demonstrate some calls with exact and symbolic input data:

>> arcsinh(1), arccosh(1/sqrt(2)), arctanh(5 + I), arccsch(1/3), 
   arcsech(I), arccoth(2)
                         /  1/2 \
                         | 2    |
      arcsinh(1), arccosh| ---- |, arctanh(5 + I), arcsinh(3),
                         \  2   /
      
         arccosh(- I), arccoth(2)
>> arcsinh(-x), arccosh(x + 1), arctanh(1/x)
                                                     / 1 \
                 -arcsinh(x), arccosh(x + 1), arctanh| - |
                                                     \ x /

Floating point values are computed for floating point arguments:

>> arcsinh(0.1234), arccosh(5.6 + 7.8*I), arccoth(1.0/10^20)
        0.1230889466, 2.956002937 + 0.9506879769 I, - 1.570796327 I

Example 2

The inverse hyperbolic functions can be rewritten in terms of the logarithm function:

>> rewrite(arcsinh(x), ln), rewrite(arctanh(x), ln)
                         2     1/2   ln(x + 1)   ln(1 - x)
                ln(x + (x  + 1)   ), --------- - ---------
                                         2           2

Example 3

The values jump when crossing a branch cut:

>> arctanh(2.0 + I/10^10), arctanh(2.0 - I/10^10)
        0.5493061443 + 1.570796327 I, 0.5493061443 - 1.570796327 I

On the branch cut, the values of arctanh coincide with the limit ``from below'' for real arguments x>1. The values coincide with the limit ``from above'' for real x<-1:

>> arctanh(1.2), arctanh(1.2 - I/10^10), arctanh(1.2 + I/10^10)
      1.198947636 - 1.570796327 I, 1.198947636 - 1.570796327 I,
      
         1.198947636 + 1.570796327 I
>> arctanh(-1.2), arctanh(-1.2 + I/10^10), arctanh(-1.2 - I/10^10)
      - 1.198947636 + 1.570796327 I, - 1.198947636 + 1.570796327 I,
      
         - 1.198947636 - 1.570796327 I

Example 4

Various system functions such as diff, float, limit, or series handle expressions involving the inverse hyperbolic functions:

>> diff(arcsinh(x^2), x), float(arccosh(3)*arctanh(5 + I))
                     2 x
                 -----------, 0.3427241326 + 2.698556745 I
                   4     1/2
                 (x  + 1)
>> limit(arcsinh(x)/arctanh(x), x = 0)
                                     1
>> series(arctanh(sinh(x)) - arcsinh(tanh(x)), x = 0, 10)
                                 7      9
                         3   83 x    4 x       10
                        x  + ----- - ---- + O(x  )
                              120    189

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