# arcsech

Inverse of the hyperbolic secant function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

arcsech(x)

## Description

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, $-\pi <\Im \left(\mathrm{arccosh}\left(x\right)\right)\le \pi$.

The inverse hyperbolic secant function is implemented according to the following relation to the logarithm function: arcsech(x) = ln(1/x + sqrt(1/x^2 - 1)). 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.

## Examples

### Example 1

Call arcsech with the following exact and symbolic input arguments:

arcsech(1), arcsech(1/sqrt(3)), arcsech(5 + I),
arcsech(1/3), arcsech(I), arcsech(2)

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

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

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

Floating-point intervals are computed for interval arguments:

arcsech(0.5...1), arcsech(0.1234...0.12345)

### Example 2

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

rewrite(arcsech(x), ln)

### Example 3

The values jump when crossing a branch cut:

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

On the branch cut, the values of arcsech coincide with the limit "from below" for real arguments x > 1:

limit(arcsech(2.0 - I/n), n = infinity);
limit(arcsech(2.0 + I/n), n = infinity);
arcsech(2.0)

The values coincide with the limit "from above" for real x < 0:

limit(arcsech(-2.0 - I/n), n = infinity);
limit(arcsech(-2.0 + I/n), n = infinity);
arcsech(-2.0)

### Example 4

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

diff(arcsech(x), x), float(arcsech(3)*arctanh(5 + I))

limit(x/arcsech(x), x = 0)

series(arcsech(x), x = 0, 3)

 x

## Return Values

Arithmetical expression or floating-point interval

x