# coth

Symbolic hyperbolic cotangent function

## Syntax

``coth(X)``

## Description

example

````coth(X)` returns the hyperbolic cotangent function of `X````

## Examples

### Hyperbolic Cotangent Function for Numeric and Symbolic Arguments

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

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

`A = coth([-2, -pi*i/3, pi*i/6, 5*pi*i/7, 3*pi*i/2])`
```A = -1.0373 + 0.0000i 0.0000 + 0.5774i 0.0000 - 1.7321i... 0.0000 + 0.7975i 0.0000 - 0.0000i```

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

`symA = coth(sym([-2, -pi*i/3, pi*i/6, 5*pi*i/7, 3*pi*i/2]))`
```symA = [ -coth(2), (3^(1/2)*1i)/3, -3^(1/2)*1i, -coth((pi*2i)/7), 0]```

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

`vpa(symA)`
```ans = [ -1.0373147207275480958778097647678,... 0.57735026918962576450914878050196i,... -1.7320508075688772935274463415059i,... 0.79747338888240396141568825421443i,... 0]```

### Plot Hyperbolic Cotangent Function

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

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

### Handle Expressions Containing Hyperbolic Cotangent Function

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

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

```syms x diff(coth(x), x) diff(coth(x), x, x)```
```ans = 1 - coth(x)^2 ans = 2*coth(x)*(coth(x)^2 - 1)```

Find the indefinite integral of the hyperbolic cotangent function:

`int(coth(x), x)`
```ans = log(sinh(x))```

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

`taylor(coth(x), x, pi*i/2)`
```ans = x - (pi*1i)/2 - (x - (pi*1i)/2)^3/3 + (2*(x - (pi*1i)/2)^5)/15```

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

`rewrite(coth(x), 'exp')`
```ans = (exp(2*x) + 1)/(exp(2*x) - 1)```

## 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.

Introduced before R2006a

## Support

#### Mathematical Modeling with Symbolic Math Toolbox

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