Inverse of the hyperbolic secant function

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.




arcsech(x) represents the inverse of the hyperbolic secant function.

arcsech 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 secant function is multivalued. MuPAD® rewrites arcsech as arcsech(x) = arccosh(1/x). The MuPAD implementation for arccosh returns values on the main branch defined by the following restriction of the imaginary part. For any finite complex x,


The inverse hyperbolic secant function is implemented according to the following relation to the logarithm function: arcsech(x) = arccosh(1/x). See Example 2.

Consequently, the branch cuts are the real intervals (-∞, 0) and (1, ∞) together with 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, arcsech is sensitive to the environment variable DIGITS which determines the numerical working precision.


Example 1

We demonstrate some calls with exact and symbolic input data:

arcsinh(1), arccosh(1/sqrt(3)), arctanh(5 + I), arccsch(1/3), 
arcsech(I), arccoth(2)

arcsinh(-x), arccosh(x + 1), arctanh(1/x)

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

arcsinh(0.1234), arccosh(5.6 + 7.8*I), arccoth(1.0/10^20)

Floating-point intervals are returned for arguments of this type:

arccoth(0.5 ... 1.5), arcsinh(0.1234...0.12345)

The inverse of the hyperbolic tangent function has real values only in the interval (- 1, 1):

arctanh(-1/2...0), arctanh(2...3)

Example 2

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

rewrite(arcsinh(x), ln), rewrite(arctanh(x), ln)

Example 3

The values jump when crossing a branch cut:

arctanh(2.0 + I/10^10), arctanh(2.0 - I/10^10)

On the branch cut, the values of arctanh coincide with the limit "from below" for real arguments x > 1. The values coincide with the limit "from above" for real x < - 1:

arctanh(1.2), arctanh(1.2 - I/10^10), arctanh(1.2 + I/10^10)

arctanh(-1.2), arctanh(-1.2 + I/10^10), arctanh(-1.2 - I/10^10)

Example 4

Various system functions such as diff, float, limit, or series handle expressions involving the inverse hyperbolic functions:

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

limit(arcsinh(x)/arctanh(x), x = 0)

series(arctanh(sinh(x)) - arcsinh(tanh(x)), x = 0, 10)

Return Values

Arithmetical expression or floating-point interval

Overloaded By


See Also

MuPAD Functions

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