sinh, cosh, tanh, csch, sech,
coth
-- the hyperbolic functionssinh(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)
.
sinh(x)
cosh(x)
tanh(x)
csch(x)
sech(x)
coth(x)
x |
- | an arithmetical expression |
an arithmetical expression.
x
When called with a floating point argument, the functions are
sensitive to the environment variable DIGITS
which determines the numerical
working precision.
arcsinh
, arccosh
, arctanh
, arccsch
, arcsech
, arccoth
Type::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.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
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 /
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
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 /
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
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
rewrite
was enhanced. Float
evaluation of tanh
and coth
for large
argument was protected against numerical overflow/underflow.