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sinh, cosh, tanh, csch, sech, coth -- the hyperbolic functions

Introduction

sinh(x) represents the hyperbolic sine function.

cosh(x) represents the hyperbolic cosine function.

tanh(x) represents the hyperbolic tangent function sinh(x)/cosh(x).

csch(x) represents the hyperbolic cosecant function 1/sinh(x).

sech(x) represents the hyperbolic secant function 1/cosh(x).

coth(x) represents the hyperbolic cotangent function cosh(x)/sinh(x).

Call(s)

sinh(x)
cosh(x)
tanh(x)
csch(x)
sech(x)
coth(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

arcsinh, arccosh, arctanh, arccsch, arcsech, arccoth

Details

Example 1

We demonstrate some calls with exact and symbolic input data:

>> sinh(I*PI), cosh(1), tanh(5 + I), csch(PI), sech(1/11), coth(8)
                                      1          1
          0, cosh(1), tanh(5 + I), --------, ----------, coth(8)
                                   sinh(PI)  cosh(1/11)
>> sinh(x), cosh(x + I*PI), tan(x^2 - 4)
                                                 2
                   sinh(x), cosh(x + I PI), tan(x  - 4)

Floating point values are computed for floating point arguments:

>> sinh(123.4), cosh(5.6 + 7.8*I), coth(1.0/10^20)
            1.953930316e53, 7.295585032 + 135.0143985 I, 1.0e20

Example 2

Simplifications are implemented for arguments that are integer multiples of I*PI/2:

>> sinh(I*PI/2), cosh(40*I*PI), tanh(-10^100*I*PI), 
   coth(-17/2*I*PI)
                                I, 1, 0, 0

Negative real numerical factors in the argument are rewritten via symmetry relations:

>> sinh(-5), cosh(-3/2*x), tanh(-x*PI/12), coth(-12/17*x*y*PI)
                      / 3 x \        / x PI \        / 12 x y PI \
        -sinh(5), cosh| --- |, - tanh| ---- |, - coth| --------- |
                      \  2  /        \  12  /        \    17     /

Example 3

The expand function implements the addition theorems:

>> expand(sinh(x + PI*I)), expand(cosh(x + y))
                -sinh(x), cosh(x) cosh(y) + sinh(x) sinh(y)

The combine function uses these theorems in the other direction, trying to rewrite products of hyperbolic functions:

>> combine(sinh(x)*sinh(y), sinhcosh)
                         cosh(x + y)   cosh(x - y)
                         ----------- - -----------
                              2             2

Example 4

Various relations exist between the hyperbolic functions:

>> csch(x), sech(x)
                                1        1
                             -------, -------
                             sinh(x)  cosh(x)

The function expand rewrites all functions in terms of sinh and cosh:

>> expand(tanh(x)), expand(coth(x))
                             sinh(x)  cosh(x)
                             -------, -------
                             cosh(x)  sinh(x)

Use rewrite to obtain a representation in terms of a specific target function:

>> rewrite(tanh(x)*exp(2*x), sinhcosh), rewrite(sinh(x), tanh)
                                                      / x \
                                                2 tanh| - |
              sinh(x) (cosh(2 x) + sinh(2 x))         \ 2 /
              -------------------------------, --------------
                          cosh(x)                      / x \2
                                               1 - tanh| - |
                                                       \ 2 /
>> rewrite(sinh(x)*coth(y), exp), rewrite(exp(x), coth)
                    2      / exp(x)   exp(-x) \      / x \
             (exp(y)  + 1) | ------ - ------- |  coth| - | + 1
                           \   2         2    /      \ 2 /
             ----------------------------------, -------------
                              2                      / x \
                        exp(y)  - 1              coth| - | - 1
                                                     \ 2 /

Example 5

The inverse functions are implemented by arcsinh, arccosh etc.:

>> sinh(arcsinh(x)), sinh(arccosh(x)), cosh(arctanh(x))
                        2     1/2            1
                   x, (x  - 1)   , ---------------------
                                          1/2        1/2
                                   (x + 1)    (1 - x)

Note that arcsinh(sinh(x)) does not necessarily yield x, because arcsinh produces values with imaginary parts in the interval [-PI/2, PI/2]:

>> arcsinh(sinh(3)), arcsinh(sinh(1.6 + 100*I))
                          3, 1.6 - 0.5309649149 I

Example 6

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

>> diff(sinh(x^2), x), float(sinh(3)*coth(5 + I))
                         2
               2 x cosh(x ), 10.01749636 - 0.0008270853591 I
>> limit(x*sinh(x)/tanh(x^2), x = 0)
                                     1
>> series((tanh(sinh(x)) - sinh(tanh(x)))/sinh(x^7), x = 0, 10)
                                       2
                                   29 x       3
                          - 1/30 + ----- + O(x )
                                    756

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