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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 iπ2 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, floating-point evaluation is fast.

Environment Interactions

When called with a floating-point argument, the functions are sensitive to the environment variable DIGITS which determines the numerical working precision.


Example 1

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)

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:

sinh(-1...1), sinh(0...1/2)

Example 2

Simplifications are implemented for arguments that are integer multiples of iπ2:

assume(n in Z_)

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)

Example 3

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)

Example 4

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)

Example 5

The inverse function is implemented as arcsinh:


Note that arcsinh(sinh(x)) does not necessarily yield xbecause arcsinh produces values with imaginary parts in the interval [π2,π2]:

arcsinh(sinh(3)), arcsinh(sinh(1.6 + 100*I))

Example 6

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)

Return Values

Arithmetical expression or a floating-point interval

Overloaded By


See Also

MuPAD Functions

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