The hyperbolic cotangent is defined as
 |
(1)
|
The notation 
is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). It is implemented
in the Wolfram Language as Coth[z].
The hyperbolic cotangent satisfies the identity
 |
(2)
|
where 
is the hyperbolic cosecant.
It has a unique real fixed point where
 |
(3)
|
at 
(OEIS A085984), which is related to the Laplace limit in the solution of Kepler's
equation.
The derivative is given by
 |
(4)
|
where 
is the hyperbolic cosecant, and the indefinite
integral by
 |
(5)
|
where 
is a constant of integration.
The Laurent series of 
is given by
(OEIS A002431 and A036278), where 
is a Bernoulli number and 
is a Bernoulli polynomial.
An asymptotic series about infinity on the real
line is given by
 |
(8)
|
See also
Bernoulli Number,
Bipolar Coordinates,
Bipolar Cylindrical
Coordinates,
Cotangent,
Hyperbolic
Functions,
Hyperbolic Tangent,
Inverse
Hyperbolic Cotangent,
Laplace's
Equation--Toroidal Coordinates,
Lebesgue Constants,
Prolate Spheroidal Coordinates,
Surface of Revolution,
Toroidal
Coordinates,
Toroidal Function
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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.Sloane, N. J. A. Sequences A002431/M0124
and A036278 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.Zwillinger,
D. (Ed.). "Hyperbolic Functions." §6.7 in CRC
Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 476-481
1995.Sloane, N. J. A. Sequences A010050
and A085984 in "The On-Line Encyclopedia
of Integer Sequences."Referenced on Wolfram|Alpha
Hyperbolic Cotangent
Cite this as:
Weisstein, Eric W. "Hyperbolic Cotangent."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCotangent.html
Subject classifications
