# acsch

Symbolic inverse hyperbolic cosecant function

## Syntax

``acsch(X)``

## Description

example

````acsch(X)` returns the inverse hyperbolic cosecant function of `X`.```

## Examples

### Inverse Hyperbolic Cosecant Function for Numeric and Symbolic Arguments

Depending on its arguments, `acsch` returns floating-point or exact symbolic results.

Compute the inverse hyperbolic cosecant function for these numbers. Because these numbers are not symbolic objects, `acsch` returns floating-point results.

`A = acsch([-2*i, 0, 2*i/sqrt(3), 1/2, i, 3])`
```A = 0.0000 + 0.5236i Inf + 0.0000i 0.0000 - 1.0472i... 1.4436 + 0.0000i 0.0000 - 1.5708i 0.3275 + 0.0000i```

Compute the inverse hyperbolic cosecant function for the numbers converted to symbolic objects. For many symbolic (exact) numbers, `acsch` returns unresolved symbolic calls.

`symA = acsch(sym([-2*i, 0, 2*i/sqrt(3), 1/2, i, 3]))`
```symA = [ (pi*1i)/6, Inf, -(pi*1i)/3, asinh(2), -(pi*1i)/2, asinh(1/3)]```

Use `vpa` to approximate symbolic results with floating-point numbers:

`vpa(symA)`
```ans = [ 0.52359877559829887307710723054658i,... Inf,... -1.0471975511965977461542144610932i,... 1.4436354751788103424932767402731,... -1.5707963267948966192313216916398i,... 0.32745015023725844332253525998826]```

### Plot Inverse Hyperbolic Cosecant Function

Plot the inverse hyperbolic cosecant function on the interval from -10 to 10.

```syms x fplot(acsch(x),[-10 10]) grid on``` ### Handle Expressions Containing Inverse Hyperbolic Cosecant Function

Many functions, such as `diff`, `int`, `taylor`, and `rewrite`, can handle expressions containing `acsch`.

Find the first and second derivatives of the inverse hyperbolic cosecant function:

```syms x diff(acsch(x), x) diff(acsch(x), x, x)```
```ans = -1/(x^2*(1/x^2 + 1)^(1/2)) ans = 2/(x^3*(1/x^2 + 1)^(1/2)) - 1/(x^5*(1/x^2 + 1)^(3/2))```

Find the indefinite integral of the inverse hyperbolic cosecant function:

`int(acsch(x), x)`
```ans = x*asinh(1/x) + asinh(x)*sign(x)```

Find the Taylor series expansion of `acsch(x)` around ```x = Inf```:

`taylor(acsch(x), x, Inf)`
```ans = 1/x - 1/(6*x^3) + 3/(40*x^5)```

Rewrite the inverse hyperbolic cosecant function in terms of the natural logarithm:

`rewrite(acsch(x), 'log')`
```ans = log((1/x^2 + 1)^(1/2) + 1/x)```

## Input Arguments

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Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.