atanh

Symbolic inverse hyperbolic tangent function

Syntax

atanh(X)

Description

atanh(X) returns the inverse hyperbolic tangent function of X.

Examples

Inverse Hyperbolic Tangent Function for Numeric and Symbolic Arguments

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

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

A = atanh([-i, 0, 1/6, i/2, i, 2])

A =
   0.0000 - 0.7854i   0.0000 + 0.0000i   0.1682 + 0.0000i...
   0.0000 + 0.4636i   0.0000 + 0.7854i   0.5493 + 1.5708i

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

symA = atanh(sym([-i, 0, 1/6, i/2, i, 2]))

symA =
[ -(pi*1i)/4, 0, atanh(1/6), atanh(1i/2), (pi*1i)/4, atanh(2)]

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

vpa(symA)

ans =
[ -0.78539816339744830961566084581988i,...
0,...
0.1682361183106064652522967051085,...
0.46364760900080611621425623146121i,...
0.78539816339744830961566084581988i,...
0.54930614433405484569762261846126 - 1.5707963267948966192313216916398i]

Plot Inverse Hyperbolic Tangent Function

Plot the inverse hyperbolic tangent function on the interval from -1 to 1.

syms x
fplot(atanh(x),[-1 1])
grid on

Handle Expressions Containing Inverse Hyperbolic Tangent Function

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

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

syms x
diff(atanh(x), x)
diff(atanh(x), x, x)

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

Find the indefinite integral of the inverse hyperbolic tangent function:

int(atanh(x), x)

ans =
log(x^2 - 1)/2 + x*atanh(x)

Find the Taylor series expansion of atanh(x):

taylor(atanh(x), x)

ans =
x^5/5 + x^3/3 + x

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

rewrite(atanh(x), 'log')

ans =
log(x + 1)/2 - log(1 - x)/2

Input Arguments

X — Input

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix

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