Taylor series expansions of hyperbolic functions, i.e., sinh, cosh, tanh, coth, sech, and csch. Home. Calculators Forum Magazines Search Members Membership Login. Series: Constants: Taylor Series Exponential Functions Logarithmic Functions Trigonometric Functions Inverse Trigonometric: Hyperbolic Functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.

Series representations. Generalized power series. Expansions at z==z 0. For the function itself

Jan 18, 2015 · $$\coth(x) = \frac{1}{x} \left ( x \coth(x) \right )$$ and then work out the Maclaurin series of $x \coth(x)$ by directly calculating derivatives. Of course this, too, quickly becomes a complicated mess as the number of terms grows.

Feb 10, 2025 · The Power Series Expansion for Hyperbolic Cotangent Function can also be presented in the form: \(\ds \coth x\) \(\ds \dfrac 1 x + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1}2^{2 n} {B_n}^* x^{2 n - 1} } {\paren {2 n}!}\)

Feb 10, 2025 · The (real) cotangent function has a Taylor series expansion: \(\ds \cot x\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}\)

I need some help to prove that the power series of $\coth x$ is: $$\frac {1} {x} + \frac {x} {3} - \frac {x^3} {45} + O (x^5) \ \ \ \ \ $$ I don't know how to derive this, should I divide the expansion ...

5 days ago · 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 expansion can be derived from the complex Fourier series of the $2 \pi$-periodic function $f(x) = e^{ax}$ for $- \pi < x < \pi$. By definition of the complex Fourier series, $$e^{ax} = \lim_{N \to \infty}\sum_{n=-N}^{N} c_{n} e^{inx} $$

Feb 9, 2018 · Both series converge (http://planetmath.org/AbsoluteConvergence) and the functions for all real (and complex) values of x x. Comparing the expansions (1) and (2) with the corresponding ones of the circular functions cosine and sine, one sees easily that. x.