# tanh

Symbolic hyperbolic tangent function

## Description

example

tanh(X) returns the hyperbolic tangent function of X.

## Examples

### Hyperbolic Tangent Function for Numeric and Symbolic Arguments

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

Compute the hyperbolic tangent function for these numbers. Because these numbers are not symbolic objects, tanh returns floating-point results.

A = tanh([-2, -pi*i, pi*i/6, pi*i/3, 5*pi*i/7])
A =
-0.9640 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.5774i...
0.0000 + 1.7321i   0.0000 - 1.2540i

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

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

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

vpa(symA)
ans =
[ -0.96402758007581688394641372410092,...
0,...
0.57735026918962576450914878050196i,...
1.7320508075688772935274463415059i,...
-1.2539603376627038375709109783365i]

### Plot Hyperbolic Tangent Function

Plot the hyperbolic tangent function on the interval from $-\pi$ to $\pi$.

syms x
fplot(tanh(x),[-pi pi])
grid on

### Handle Expressions Containing Hyperbolic Tangent Function

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

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

syms x
diff(tanh(x), x)
diff(tanh(x), x, x)
ans =
1 - tanh(x)^2

ans =
2*tanh(x)*(tanh(x)^2 - 1)

Find the indefinite integral of the hyperbolic tangent function:

int(tanh(x), x)
ans =
log(cosh(x))

Find the Taylor series expansion of tanh(x):

taylor(tanh(x), x)
ans =
(2*x^5)/15 - x^3/3 + x

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

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

## 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.