双曲三角函数 与三角函数是数学上最为常见的运算公式，这篇文章主要讲述一些关于双曲三角函数以及三角函数的 泰勒展开式。 双曲三角函数,我们由定义很容易知道 \sinh {x},\cosh {x} 的展开 …

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 …

Taylor series expansions of hyperbolic functions, i.e., sinh, cosh, tanh, coth, sech, and csch.

2015年1月18日 · There is a simple way of approximating coth by noticing that it is a logarithmic derivative. Since: sinhz z = + ∞ ∏ n = 1(1 + z2 π2n2) by the Weierstrass product for the …

Series representations Generalized power series Expansions at z == z0 For the function itself Series representations (27 formulas)

2025年2月10日 · Hence by the Ratio Test, the series converges for |x| <π. The Power Series Expansion for Hyperbolic Cotangent Function can also be presented in the form: 1 x + ∞ ∑ n = …

2017年10月2日 · I recently encountered this series representation of hyperbolic cotangent function. How this equation can be derived? $$\coth (z) =\sum_ {k=-\infty}^ {\infty}\frac {z} {z^ …

The expansion can be derived from the complex Fourier series of the 2π 2 π -periodic function f(x) =eax f (x) = e a x for −π <x <π − π <x <π. By definition of the complex Fourier series,

2025年3月5日 · 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 …

2021年2月5日 · We know that the coefficients for this power series are the Bernoulli numbers. This result is important because it allows us to arrive at the power series of $\cot (x)$ and …