Hyperbolic sine function
This functionality does not run in MATLAB.
sinh(x
)
sinh(x)
represents the hyperbolic sine function.
This function is defined for complex arguments.
Floatingpoint values are returned for floatingpoint arguments. Floatingpoint intervals are returned for floatingpoint interval arguments. Unevaluated function calls are returned for most exact arguments.
Arguments that are integer multiples of $$\frac{i\pi}{2}$$ lead
to simplified results. If the argument involves a negative numerical
factor of Type::Real
,
then symmetry relations are used to make this factor positive. See Example 2.
The special values sinh(0) = 0
, sinh(∞)
= ∞
, and sinh(–∞) = –∞
are
implemented.
The functions expand
and combine
implement the
addition theorems for the hyperbolic functions. See Example 3.
You can rewrite other hyperbolic functions in terms of sinh
and cosh
.
For example, csch(x)
is rewritten as 1/sinh(x)
.
Use expand
or rewrite
to rewrite expressions
involving tanh
and coth
in terms
of sinh
and cosh
. See Example 4.
The inverse function is implemented by arcsinh
.
See Example 5.
The float attributes are kernel functions, thus, floatingpoint evaluation is fast.
When called with a floatingpoint argument, the functions are
sensitive to the environment variable DIGITS
which determines
the numerical working precision.
Call sinh
with the following exact and symbolic
input arguments:
sinh(I*PI), sinh(1), sinh(5 + I), sinh(PI), sinh(1/11), sinh(8)
sinh(x), sinh(x + I*PI), sinh(x^2  4)
Floatingpoint values are computed for floatingpoint arguments:
sinh(1.234), sinh(5.6 + 7.8*I), sinh(1.0/10^20)
Floatingpoint intervals are computed for interval arguments:
sinh(1...1), sinh(0...1/2)
Simplifications are implemented for arguments that are integer multiples of $$\frac{i\pi}{2}$$:
assume(n in Z_)
simplify(sinh(n*I*PI))
simplify(sinh((n  1/2)*I*PI))
delete n
Negative real numerical factors in the argument are rewritten via symmetry relations:
sinh(5), sinh(3/2*x), sinh(x*PI/12), sinh(12/17*x*y*PI)
The expand
function
implements the addition theorems:
expand(sinh(x + PI*I)), expand(sinh(x + y))
The combine
function
uses these theorems in the other direction, trying to rewrite products
of hyperbolic functions:
combine(sinh(x)*sinh(y), sinhcosh)
Use rewrite
to
obtain a representation in terms of a specific target function:
rewrite(sinh(x)*exp(2*x), sinhcosh); rewrite(sinh(x), tanh)
rewrite(sinh(x)*coth(y), exp); rewrite(exp(x), sinhcosh)
The inverse function is implemented as arcsinh
:
sinh(arcsinh(x)), arcsinh(sinh(x))
Note that arcsinh(sinh(x))
does not necessarily
yield x
because arcsinh
produces
values with imaginary parts in the interval $$\left[\frac{\pi}{2},\frac{\pi}{2}\right]$$:
arcsinh(sinh(3)), arcsinh(sinh(1.6 + 100*I))
diff
, float
, limit
, taylor
, series
and other system
functions handle expressions involving the hyperbolic functions:
diff(sinh(x^2), x), float(sinh(3)*coth(5 + I))
limit(x*sinh(x)/tanh(x^2), x = 0)
taylor(sinh(x), x = 0)
series((tanh(sinh(x))  sinh(tanh(x)))/sinh(x^7), x = 0)

Arithmetical expression or a floatingpoint interval
x