# csch

Symbolic hyperbolic cosecant function

## Syntax

``csch(X)``

## Description

example

````csch(X)` returns the hyperbolic cosecant function of `X`.```

## Examples

### Hyperbolic Cosecant Function for Numeric and Symbolic Arguments

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

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

`A = csch([-2, -pi*i/2, 0, pi*i/3, 5*pi*i/7, pi*i/2])`
```A = -0.2757 + 0.0000i 0.0000 + 1.0000i Inf + 0.0000i... 0.0000 - 1.1547i 0.0000 - 1.2790i 0.0000 - 1.0000i```

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

`symA = csch(sym([-2, -pi*i/2, 0, pi*i/3, 5*pi*i/7, pi*i/2]))`
```symA = [ -1/sinh(2), 1i, Inf, -(3^(1/2)*2i)/3, 1/sinh((pi*2i)/7), -1i]```

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

`vpa(symA)`
```ans = [ -0.27572056477178320775835148216303,... 1.0i,... Inf,... -1.1547005383792515290182975610039i,... -1.2790480076899326057478506072714i,... -1.0i]```

### Plot Hyperbolic Cosecant Function

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

```syms x fplot(csch(x),[-10 10]) grid on```

### Handle Expressions Containing Hyperbolic Cosecant Function

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

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

```syms x diff(csch(x), x) diff(csch(x), x, x)```
```ans = -cosh(x)/sinh(x)^2 ans = (2*cosh(x)^2)/sinh(x)^3 - 1/sinh(x)```

Find the indefinite integral of the hyperbolic cosecant function:

`int(csch(x), x)`
```ans = log(tanh(x/2))```

Find the Taylor series expansion of `csch(x)` around ```x = pi*i/2```:

`taylor(csch(x), x, pi*i/2)`
```ans = ((x - (pi*1i)/2)^2*1i)/2 - ((x - (pi*1i)/2)^4*5i)/24 - 1i```

Rewrite the hyperbolic cosecant function in terms of the exponential function:

`rewrite(csch(x), 'exp')`
```ans = -1/(exp(-x)/2 - exp(x)/2)```

## Input Arguments

collapse all

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

## Version History

Introduced before R2006a