
双曲三角函数与三角函数泰勒展开式 - 知乎 - 知乎专栏
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} 的展开式: \sinh {x}=\frac {e^ {n}-e^ {-n}} {2} =\frac {1} {2}\sum\limits_ {n=0}^ {\infty}\left (\frac {x^ {n}} {n!}-\frac { (-x)^ {n}} {n!}\right) =\sum\limits_ {n=0}^ {\infty}\frac {x^ {2n+1}} { (2n+1)!},x\in { (-\infty,+\infty)}
Hyperbolic functions - Wikipedia
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 - eFunda
Taylor series expansions of hyperbolic functions, i.e., sinh, cosh, tanh, coth, sech, and csch.
taylor expansion - Approximate $\coth (x)$ around $x = 0
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 (hyperbolic) sine function, we have: logsinhz − logz = + ∞ ∑ n = 1log(1 + z2 π2n2), so, by differentiating both sides: cothz − 1 z = + ∞ ∑ n = 1 2z π2n2 + z2.
Hyperbolic cotangent: Series representations - Wolfram
Series representations Generalized power series Expansions at z == z0 For the function itself Series representations (27 formulas)
Power Series Expansion for Hyperbolic Cotangent Function
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 = 1(− 1)n − 122nBn ∗ x2n − 1 (2n)! where Bn ∗ denotes the archaic form of the Bernoulli numbers.
Series representation of hyperbolic cotangent
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^ …
Series expansion of $\\coth x$ using the Fourier transform
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,
Hyperbolic Cotangent -- from Wolfram MathWorld
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 u^*=1.19967874...
Deriving the power series for $\cot (x)$ and $\coth (x)$
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 $\coth (x)$. In this post, we will see how the Bernoulli numbers are related to the expansion of $\cot (x)$ and $\coth (x)$. (added in Feb, 11, 2021):
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