Hyperbolic sine function
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sinh(x) represents the hyperbolic sine function.
This function is defined for complex arguments.
Floating-point values are returned for floating-point arguments. Floating-point intervals are returned for floating-point interval arguments. Unevaluated function calls are returned for most exact arguments.
Arguments that are integer multiples of lead
to simplified results. If the argument involves a negative numerical
then symmetry relations are used to make this factor positive. See Example 2.
The special values
sinh(0) = 0,
= ∞, and
sinh(–∞) = –∞ are
You can rewrite other hyperbolic functions in terms of
csch(x) is rewritten as
rewrite to rewrite expressions
coth in terms
cosh. See Example 4.
The inverse function is implemented by
See Example 5.
The float attributes are kernel functions, thus, floating-point evaluation is fast.
When called with a floating-point argument, the functions are
sensitive to the environment variable
DIGITS which determines
the numerical working precision.
sinh with the following exact and symbolic
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)
Floating-point values are computed for floating-point arguments:
sinh(1.234), sinh(5.6 + 7.8*I), sinh(1.0/10^20)
Floating-point intervals are computed for interval arguments:
Simplifications are implemented for arguments that are integer multiples of :
assume(n in Z_)
simplify(sinh((n - 1/2)*I*PI))
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)
implements the addition theorems:
expand(sinh(x + PI*I)), expand(sinh(x + y))
uses these theorems in the other direction, trying to rewrite products
of hyperbolic functions:
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(x)) does not necessarily
values with imaginary parts in the interval :
arcsinh(sinh(3)), arcsinh(sinh(1.6 + 100*I))
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 floating-point interval