sinh, cosh, tanh, csch, sech,
coth -- the hyperbolic functions
Introductionsinh(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)
Parametersx |
- | an arithmetical expression |
Returnsan arithmetical expression.
x
Side
EffectsWhen called with a floating point argument, the functions are
sensitive to the environment variable DIGITS which determines the numerical
working precision.
Related
Functionsarcsinh, arccosh, arctanh, arccsch, arcsech, arccoth
DetailsType::Real, then
symmetry relations are used to make this factor positive. Cf.
example 2.sinh(0)=0, sinh(+/-infinity) = +/-infinity,
cosh(0)=1, cosh(+/- infinity)=infinity,
tanh(0)=1, tanh(+/-infinity)= +/-infinity,
coth(+/-infinity)= +/-infinity
are implemented.
expand and combine implement the addition
theorems for the hyperbolic functions. Cf. example 3.sech(x) and csch(x) are rewritten as
1/cosh(x) and 1/sinh(x), respectively. Use
expand or rewrite to rewrite expressions
involving tanh and coth in terms of
sinh and cosh. Cf. example 4.arcsin,
arccos, arctan, arccsc,
arcsec, and arccot, respectively. Cf.
example 5.
Example
1We 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
2Simplifications 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
3The 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
4Various 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
5The 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
6Various 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
rewrite was enhanced. Float
evaluation of tanh and coth for large
argument was protected against numerical overflow/underflow.