Hyperbolic tangent

The hyperbolic tangent is sinh divided by cosh and approaches plus or minus one at the two infinities.

Table of contents

Inverse hyperbolic tangent ( $artanh$ )

For

∣ x ∣ < 1

, the inverse hyperbolic tangent is

artanh x = \frac{1}{2} lo g \frac{1 + x}{1 - x} .

Small-argument expansion of the hyperbolic tangent ( $tanh x = x + O (x^{3})$ )

The taylor series at zero begins

tanh x = x - \frac{x ^{3}}{3} + O (x^{5}),

so

tanh x \sim x

x \to 0

.

Hyperbolic-function identity

The exponential definitions of the hyperbolic functions imply identities such as

\frac{s i n h x}{1 + c o s h x} = tanh (\frac{x}{2}) .

Hyperbolic Pythagorean identity

The identity

cosh^{2} x - sinh^{2} x = 1

implies

1 - tanh^{2} x = sech^{2} x, coth^{2} x - csch^{2} x = 1.

 Ancestors (6)

 Incoming links (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version