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Hyperbolic Cotangent


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The hyperbolic cotangent is defined as

 cothz=(e^z+e^(-z))/(e^z-e^(-z))=(e^(2z)+1)/(e^(2z)-1).
(1)

The notation cthz 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

 coth(z/2)-cothz=cschz,
(2)

where cschz is the hyperbolic cosecant.

It has a unique real fixed point where

 cothu=u
(3)

at u^*=1.19967874... (OEIS A085984), which is related to the Laplace limit in the solution of Kepler's equation.

The derivative is given by

 d/(dz)cothz=-csch^2z,
(4)

where cschz is the hyperbolic cosecant, and the indefinite integral by

 intcothzdz=ln(sinhz)+C,
(5)

where C is a constant of integration.

The Laurent series of cothz is given by

cothz=1/z+sum_(n=1)^(infty)(2^(2n)B_(2n))/((2n)!)z^(2n-1)
(6)
=1/z+1/3z-1/(45)z^3+2/(945)z^5-...
(7)

(OEIS A002431 and A036278), where B_n is a Bernoulli number and B_n(z) is a Bernoulli polynomial. An asymptotic series about infinity on the real line is given by

 cothz∼1+2e^(-2z)+2e^(-4z)+....
(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 tanh(x) and Cotangent coth(x) 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."

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Hyperbolic Cotangent

Cite this as:

Weisstein, Eric W. "Hyperbolic Cotangent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCotangent.html

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