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Inverse of the hyperbolic cosecant function

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.




arccsch(x) represents the inverse of the hyperbolic cosecant function.

arccsch 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 inverse hyperbolic cosecant function is multivalued. MuPAD® rewrites arccsch as arccsch(x) = arcsinh(1/x). The MuPAD implementation for arcsinh returns values on the main branch defined by the following restriction of the imaginary part. For any finite complex x, π2(arcsinh(x))π2

The inverse hyperbolic cosecant function is implemented according to the following relation to the logarithm function: arccsch(x) = ln(1/x + sqrt(1/x^2 + 1)). See Example 2.

Consequently, the branch cut is the interval (-i, i) on the imaginary axis. The values jump when the argument crosses a branch cut. See Example 3.

The float attributes are kernel functions, and floating-point evaluation is fast.

Environment Interactions

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


Example 1

Call arccsch with the following exact and symbolic input arguments:

arccsch(-1), arccsch(1/sqrt(3)), arccsch(5 + I),
arccsch(1/3), arccsch(I), arccsch(2)

arccsch(-x), arccsch(x + 1), arccsch(1/x)

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

arccsch(0.1234), arccsch(5.6 + 7.8*I), arccsch(1.0/10^20)

Floating-point intervals are computed for interval arguments:

arccsch(-1.5...-0.5), arccsch(0.1234...0.12345)

Example 2

The inverse hyperbolic cosecant function can be rewritten in terms of the logarithm function:

rewrite(arccsch(x), ln)

Example 3

The values jump when crossing a branch cut:

arccsch(0.5*I + 1/10^10), arccsch(0.5*I - 1/10^10)

On the branch cut, the values of arccsch coincide with the limit “from the left” for imaginary arguments x = c*i where -1 < c < 0:

limit(arccsch(0.5*I - 1/n), n = infinity);
limit(arccsch(0.5*I + 1/n), n = infinity);

The values coincide with the limit “from the right” for imaginary arguments x = c*i where 0 < c < 1:

limit(arccsch(-0.5*I - 1/n), n = infinity);
limit(arccsch(-0.5*I + 1/n), n = infinity);

Example 4

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

diff(arccsch(x^2), x), float(arccsch(3)*arctanh(5 + I))

limit(x/arccsch(1/x), x = 0)
taylor(arccsch(1/x), x = 0)

series(arccsch(x), x = 0, Right)

Return Values

Arithmetical expression or floating-point interval

Overloaded By


See Also

MuPAD Functions

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