$\text{tanh} (x + k\pi i) = \text{tanh}\ x$ $\text{coth} (x + k\pi i) =\text{coth} x$ Relationship between inverse hyperbolic and inverse trigonometric 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.

在双曲函数中，coth为双曲余切。 在数学中，双曲余切是由基础双曲函数双曲正弦和双曲余弦推导而来。 双曲余切函数是双曲正切函数的倒数。

coth (x) = cosh (x) sinh (x) = ex + e-x ex − e-x. sech (x) = 1 cosh (x) = 2 ex + e-x. csch (x) = 1 sinh (x) = 2 ex − e-x. Why the Word "Hyperbolic" ? Because it comes from measurements made on a Hyperbola: So, just like the trigonometric functions relate to a circle, the hyperbolic functions relate to a hyperbola. And.

Illustrated definition of Coth: The Hyperbolic Cotangent Function. coth (x) (esupxsup esupminusxsup) (esupxsup minus...

Coth is a mathematical function that is used in various areas of mathematics, including calculus, trigonometry, and algebra. It is the hyperbolic cotangent function, which is defined as the ratio of the hyperbolic cosine to the hyperbolic sine of an angle.

2021年2月6日 · The coth function is a hyperbolic function. It is also known as the hyperbolic cotangent function. Here is a video that explains the cosech, sech, and coth functions: The coth function is the reciprocal of the tanh function, and is only defined for non-zero values: coth x = 1 t a n h x = cosh x sinh x.

coth (z) = i cot (iz)

2024年11月12日 · coth函数，即双曲余切函数，定义为双曲正弦函数（sinh）与双曲余弦函数（cosh）的比值，即： coth (x) = cosh (x) / sinh (x) 这可以进一步展开为e的指数形式的比值： coth (x) = (e^x + e^-x) / (e^x - e^-x) 其中，e是自然对数的底数。 这个定义表明，coth函数在x=0时是不定义的，因为分母会变为0。 同时，当x趋于正无穷或负无穷时，coth (x)将趋于1或-1。 coth函数具有一些重要的性质。 它是奇函数，这意味着coth (-x) = -coth (x)。 此外，coth函数在整个 …

Definition 4.11.1 The hyperbolic cosine is the function coshx = ex + e − x 2, and the hyperbolic sine is the function sinhx = ex − e − x 2. . Notice that cosh is even (that is, cosh(− x) = cosh(x)) while sinh is odd (sinh(− x) = − sinh(x)), and coshx + sinhx = ex.