# coth

Hyperbolic cotangent

## Description

example

Y = coth(X) returns the hyperbolic tangent of the elements of X. The coth function operates element-wise on arrays. The function accepts both real and complex inputs. All angles are in radians.

## Examples

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Create a vector and calculate the hyperbolic cotangent of each value.

X = [0 pi 2*pi 3*pi];
Y = coth(X)
Y = 1×4

Inf    1.0037    1.0000    1.0000

Plot the hyperbolic cotangent over the domain $-\pi and $0.

x1 = -pi+0.01:0.01:-0.01;
x2 = 0.01:0.01:pi-0.01;
y1 = coth(x1);
y2 = coth(x2);
plot(x1,y1,x2,y2)
grid on

## Input Arguments

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Input angles in radians, specified as a scalar, vector, matrix, or multidimensional array.

Data Types: single | double
Complex Number Support: Yes

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### Hyperbolic Cotangent

The hyperbolic cotangent of x is equal to the inverse of the hyperbolic tangent

$\mathrm{coth}\left(x\right)=\frac{1}{\mathrm{tanh}\left(x\right)}=\frac{{e}^{2x}+1}{{e}^{2x}-1}.$

In terms of the traditional cotangent function with a complex argument, the identity is

$\mathrm{coth}\left(x\right)=i\mathrm{cot}\left(ix\right)\text{\hspace{0.17em}}.$