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# arcsech

Inverse of the hyperbolic secant function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```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,

,

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.

## Examples

### 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)`

 x

## Return Values

Arithmetical expression or floating-point interval

x

## See Also

### MuPAD Functions

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