By way of analogy with the usual tangent
the hyperbolic tangent is defined as
is the hyperbolic sine and is the hyperbolic cosine.
is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix).
is implemented in the Wolfram Language
Special values include
is the golden ratio.
The derivative of is
and higher-order derivatives are given by
is an Eulerian number.
The indefinite integral is given by
has Taylor series
(OEIS A002430 and A036279).
As Gauss showed in 1812, the hyperbolic tangent can be written using a continued
(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 second-order
ordinary differential equation
together with the boundary conditions and .
See alsoBernoulli Number
, Correlation Coefficient--Bivariate
, Fisher's z-'-Transformation
, Inverse Hyperbolic Tangent
, Oblate Spheroidal Coordinates
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ReferencesAbramowitz, 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. 83-86, 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. 117-122, 2000.Olds, C. D. Continued
Fractions. New York: Random House, 1963.Sloane, N. J. A.
Sequences A002430/M2100 and A036279
in "The On-Line 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. 279-284, 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. 476-481
Referenced on Wolfram|AlphaHyperbolic Tangent
Cite this as:
Weisstein, Eric W. "Hyperbolic Tangent."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicTangent.html