Hyperbolic Tangent
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By way of analogy with the usual tangent
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(1)
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the hyperbolic tangent is defined as
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(2)
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(3)
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(4)
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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 Mathematica
as Tanh[z].
Special values include
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(5)
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(6)
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where is the golden
ratio.
The derivative of is
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(7)
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and higher-order derivatives are given by
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(8)
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where is an Eulerian
number.
The indefinite integral is given by
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(9)
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has Taylor
series
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(10)
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(11)
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(OEIS A002430 and A036279).
As Gauss showed in 1812, the hyperbolic tangent can be written using a continued fraction as
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(12)
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(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
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(13)
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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. mathworld.wolfram.com/HyperbolicTangent.html