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`arccsc`

Inverse cosecant function

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

Syntax

```arccsc(`x`)
```

Description

`arccsc(x)` represents the inverse of the cosecant function.

The angle returned by this function is measured in radians, not in degrees. For example, the result π represents an angle of 180o.

`arccsc` is defined for complex arguments.

Floating-point values are returned for floating-point arguments. Floating-point intervals are returned for interval arguments. Unevaluated function calls are returned for most exact arguments.

If the argument is a rational multiple of `I`, the result is expressed in terms of hyperbolic functions. See Example 2.

MuPAD® rewrites `arccsc` as ```arccsc(x) = arcsin(1/x)```.

The inverse cosecant functions is multivalued. The MuPAD `arccsc` function returns values on the main branch. The branch cut is the real interval (- 1, 1). Thus, `arccsc` returns values, such that y = arccsc(x) satisfies $-\frac{\pi }{2}\le \Re \left(y\right)\le \frac{\pi }{2},\text{\hspace{0.17em}}y\ne 0$ for any finite complex x.

The `arccsc` function returns explicit values for arguments that are certain rational multiples of π. For these values, the inverse functions return an appropriate rational multiple of π on the main branch. See Example 3.

The values jump when the arguments cross a branch cut. See Example 4.

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

Environment Interactions

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

Examples

Example 1

Call `arccsc` with the following exact and symbolic input arguments:

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

`arccsc(-x), arccsc(x + 1), arccsc(1/x)`

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

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

Floating-point intervals are computed for interval arguments:

`arccsc(-2...-1), arccsc(1...5)`

Note that certain types of input lead to severe overestimation, sometimes returning the whole image set of the function in question:

```arccsc(-2...2); csc(arccsc(-2...2))```

Example 2

Arguments that are rational multiples of `I` are rewritten in terms of hyperbolic functions:

`arccsc(5*I), arccsc(5/4*I), arccsc(-3*I)`

For other complex arguments unevaluated function calls without simplifications are returned:

`arccsc(1/2^(1/2) + I), arccsc(1 - 3*I)`

Example 3

Some special values are implemented:

`arccsc(sqrt(2)), arccsc(4/(5^(1/2) - 1)), arccsc(2/3^(1/2))`

Such simplifications occur for arguments that are trigonometric images of rational multiples of π:

`csc(9/10*PI), arccsc(csc(9/10*PI))`

Example 4

The values jump when crossing a branch cut:

`arccsc(0.5 + I/10^10), arccsc(0.5 - I/10^10)`

On the branch cut, the values of `arccsc` coincide with the limit “from above” for real arguments ```0 < x < 1```:

```limit(arccsc(0.5 - I/n), n = infinity); limit(arccsc(0.5 + I/n), n = infinity); arccsc(0.5)```

The values coincide with the limit “from below” for real `-1 < x < 0`:

```limit(arccsc(-0.5 - I/n), n = infinity); limit(arccsc(-0.5 + I/n), n = infinity); arccsc(-0.5)```

Example 5

The inverse cosecant function can be rewritten in terms of the logarithm function with complex arguments:

`rewrite(arccsc(x), ln)`

Example 6

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

`diff(arccsc(x^2), x), float(arccsc(3)*arctan(5 + I))`

`limit(arccsc(1 + sin(x)/x), x = 0)`

`taylor(arccsc(1/x), x = 0)`

`series(arccsc(x), x = 0, Right)`

Parameters

 `x`

Return Values

Arithmetical expression or floating-point interval.

`x`