Hyperbolic Tangent

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By way of analogy with the usual tangent


the hyperbolic tangent is defined as


where sinhz is the hyperbolic sine and coshz is the hyperbolic cosine. The notation thz is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix).

tanhz is implemented in the Wolfram Language as Tanh[z].

Special values include


where phi is the golden ratio.

The derivative of tanhz is


and higher-order derivatives are given by

 (d^n)/(dz^n)tanhz=(2^(n+1)e^(2z))/((1+e^(2z))^(n+1))sum_(k=0)^(n-1)<n; k>(-1)^ke^(2kz),

where <n; k> is an Eulerian number.

The indefinite integral is given by


tanhz has Taylor series


(OEIS A002430 and A036279).

As Gauss showed in 1812, the hyperbolic tangent can be written using a continued fraction as


(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 tanhx satisfies the second-order ordinary differential equation


together with the boundary conditions f(0)=0 and f^'(infty)=0.

See also

Bernoulli Number, Catenary, Correlation Coefficient--Bivariate Normal Distribution, Fisher's z-'-Transformation, Hyperbolic Cotangent, Hyperbolic Functions, Inverse Hyperbolic Tangent, Lorentz Group, Mercator Projection, Oblate Spheroidal Coordinates, Pseudosphere, Surface of Revolution, Tangent, Tractrix

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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 tanh(x) and Cotangent coth(x) 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.

Referenced on Wolfram|Alpha

Hyperbolic Tangent

Cite this as:

Weisstein, Eric W. "Hyperbolic Tangent." From MathWorld--A Wolfram Web Resource.

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