The hyperbolic cosecant is defined as
|
(1)
|
It is implemented in the Wolfram Language
as Csch[z].
It is related to the hyperbolic cotangent
though
|
(2)
|
The derivative is given by
|
(3)
|
where
is the hyperbolic cotangent, and the indefinite
integral by
|
(4)
|
where
is a constant of integration.
It has Taylor series
(OEIS A036280 and A036281), where
is a Bernoulli polynomial and is a Bernoulli number.
Sums include
(OEIS A110191; Berndt 1977).
The plot above shows a bifurcation diagram for .
See also
Bernoulli Number,
Bipolar Coordinates,
Bipolar Cylindrical
Coordinates,
Cosecant,
Helmholtz
Differential Equation--Toroidal Coordinates,
Hyperbolic
Functions,
Hyperbolic Sine,
Inverse
Hyperbolic Cosecant,
Poinsot's Spirals,
Surface of Revolution,
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.Berndt, B. C. "Modular
Transformations and Generalizations of Several Formulae of Ramanujan." Rocky
Mtn. J. Math. 7, 147-189, 1977.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 A036280,
A036281, and A110191
in "The On-Line Encyclopedia of Integer Sequences."Spanier,
J. and Oldham, K. B. "The Hyperbolic Secant and Cosecant Functions." Ch. 29 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 273-278, 1987.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 Cosecant
Cite this as:
Weisstein, Eric W. "Hyperbolic Cosecant."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCosecant.html
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