By way of analogy with the usual tangent

(1)

the hyperbolic tangent is defined as
where
is the hyperbolic sine and is the hyperbolic cosine.
The notation
is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix).
is implemented in the Wolfram Language
as Tanh[z].
Special values include
where
is the golden ratio.
The derivative of is

(7)

and higherorder derivatives are given by

(8)

where
is an Eulerian number.
The indefinite integral is given by

(9)

has Taylor series
(OEIS A002430 and A036279).
As Gauss showed in 1812, the hyperbolic tangent can be written using a continued
fraction as

(12)

(Wall 1948, p. 349; Olds 1963, p. 138). This continued fraction is also known as Lambert's continued fraction
(Wall 1948, p. 349).
The hyperbolic tangent satisfies the secondorder
ordinary differential equation

(13)

together with the boundary conditions and .
See also
Bernoulli Number,
Catenary,
Correlation CoefficientBivariate
Normal Distribution,
Fisher's z'Transformation,
Hyperbolic Cotangent,
Hyperbolic
Functions,
Inverse Hyperbolic Tangent,
Lorentz Group,
Mercator
Projection,
Oblate Spheroidal Coordinates,
Pseudosphere,
Surface
of Revolution,
Tangent,
Tractrix
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References
Abramowitz, M. and Stegun, I. A. (Eds.). "Hyperbolic Functions." §4.5 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 8386, 1972.Gradshteyn, I. S. and Ryzhik,
I. M. Tables
of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
2000.Jeffrey, A. "Hyperbolic Identities." §2.5 in Handbook
of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press,
pp. 117122, 2000.Olds, C. D. Continued
Fractions. New York: Random House, 1963.Sloane, N. J. A.
Sequences A002430/M2100 and A036279
in "The OnLine Encyclopedia of Integer Sequences."Spanier,
J. and Oldham, K. B. "The Hyperbolic Tangent and Cotangent Functions." Ch. 30 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 279284, 1987.Wall,
H. S. Analytic
Theory of Continued Fractions. New York: Chelsea, 1948.Zwillinger,
D. (Ed.). "Hyperbolic Functions." §6.7 in CRC
Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 476481
1995.Referenced on WolframAlpha
Hyperbolic Tangent
Cite this as:
Weisstein, Eric W. "Hyperbolic Tangent."
From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicTangent.html
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