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Date: 10-8-2019
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By way of analogy with the usual tangent
(1) |
the hyperbolic tangent is defined as
(2) |
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(3) |
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(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) |
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(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) |
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(11) |
(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 second-order ordinary differential equation
(13) |
together with the boundary conditions and .
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. 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.
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