Hyperbolic Cotangent

The hyperbolic cotangent is defined as

$$\coth x\equiv {e^x+e^{-x}\over e^x-e^{-x}} = {e^{2x}+1\over e^{2x}-1}.$$

Its Laurent Series is

$$\coth x = {1\over x}+{\textstyle{1\over 3}} x-{\textstyle{1\over 45}} x^3+\ldots.$$

See also Bernoulli Number, Bipolar Coordinates, Bipolar Cylindrical Coordinates, Cotangent, Fibonacci Hyperbolic Cotangent, Hyperbolic Tangent, Laplace's Equation--Toroidal Coordinates, Lebesgue Constants (Fourier Series), Prolate Spheroidal Coordinates, Surface of Revolution, Toroidal Coordinates, Toroidal Function

References

Abramowitz, M. and Stegun, C. 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.

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.