Hyperbolic Tangent
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 |
By way of analogy with the usual tangent
 |
(1)
|
the hyperbolic tangent is defined as
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
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
 |  |  |
(5)
|
 |  |  |
(6)
|
where 
is the golden
ratio.
The derivative of 
is
 |
(7)
|
and higher-order derivatives are given by
 |
(8)
|
where 
is an Eulerian
number.
The indefinite integral is given by
 |
(9)
|
has Taylor
series
 |  |  |
(10)
|
 |  |  |
(11)
|
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 second-order ordinary
differential equation
 |
(13)
|
together with the boundary conditions 
and 
.
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. 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 1995.
Weisstein, Eric W. "Hyperbolic Tangent." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HyperbolicTangent.html
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inverse hyperbolic
tangent of x
