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# csch

Hyperbolic cosecant function

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csch(x)

## Description

csch(x) represents the hyperbolic cosecant function, 1/sinh(x). 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.

The hyperbolic cosecant function simplifies to ${\left(-1\right)}^{n+1/2}$ at the points $\frac{\pi i}{2}+\pi in$, where n is an integer. The hyperbolic cosecant function has singularities at the points $i\pi n$, where n is an integer. 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 functions expand and combine implement the addition theorems for the hyperbolic functions. See Example 3.

csch(x) is rewritten as 1/sinh(x). Use expand or rewrite to rewrite expressions involving csch in terms of other functions. See Example 4.

The inverse function is implemented as arccsch. 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.

## Examples

### Example 1

Call csch with the following exact and symbolic input arguments:

csch(I*PI/2), csch(1), csch(5 + I), csch(PI), csch(1/11), csch(8)

csch(x), csch(x + I*PI), csch(x^2 - 4)

Floating-point values are computed for floating-point arguments:

csch(1.234), csch(5.6 + 7.8*I), csch(1.0/10^20)

Floating-point intervals are computed for interval arguments:

csch(-1...-1/2), csch(1...10)

For functions with discontinuities, evaluation over an interval can return in a union of intervals:

csch(-1...1)

### Example 2

The hyperbolic cosecant function equals simplifies to ${\left(-1\right)}^{n+1/2}$ at the points , where n is an integer:

assume(n in Z_)
simplify(csch((n - 1/2)*I*PI))

delete n

Negative real numerical factors in the argument are rewritten via symmetry relations:

csch(-5), csch(-3/2*x), csch(-x*PI/12), csch(-12/17*x*y*PI)

### Example 3

The expand function implements the addition theorems:

expand(csch(x + PI*I)), expand(csch(x + y))

### Example 4

csch(x) is automatically rewritten as 1/sinh(x):

csch(x)

Use rewrite to obtain a representation in terms of other target functions:

rewrite(csch(x)*exp(2*x), sinhcosh), rewrite(csch(x), exp)

rewrite(csch(x)*coth(y), sincos), rewrite(csch(x), tanh)

### Example 5

The inverse function is implemented as arccsch:

csch(arccsch(x)),
arccsch(csch(x))

Note that arccsch(csch(x)) does not necessarily yield x, because arccsch produces values with imaginary parts in the interval $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$:

arccsch(csch(3)), arccsch(csch(1.6 + 100*I))

### Example 6

diff, float, limit, taylor, series, and other system functions handle expressions involving the hyperbolic functions:

diff(csch(x), x), float(csch(3)*coth(5 + I))

limit(x*csch(x)/cosh(x^2), x = 0)

taylor(1/csch(x), x = 0)

series(csch(x), x = 0)

 x

## Return Values

Arithmetical expression or a floating-point interval

x