Hyperbolic Tangent
المؤلف:
Abramowitz, M. and Stegun, I. A
المصدر:
"Hyperbolic Functions." §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover,
الجزء والصفحة:
...
3-6-2019
2058
Hyperbolic Tangent
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 higher-order 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 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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