For arcoth, the argument of the logarithm is in (−∞, 0], if and only if z belongs to the real interval [−1, 1]. Therefore, these formulas define convenient principal values, for which the branch cuts are (−∞, −1] and [1, ∞) for the inverse hyperbolic tangent, and [−1, 1] …

2020年2月3日 · Logarithm formula for artanh. There is a also a formula for finding artanh directly: $$ \operatorname{artanh} x =\frac12\ln\left(\frac{1+x}{1-x}\right) $$ Here is a proof of the logarithm formula for artanh. This follows similar lines to the proof for arsinh. We will use: $$ u = \operatorname{artanh}{x} $$

2022年9月2日 · The hyperbolic functions have “h”, but the “h” doesn’t come immediately after the “ar”: arsinh x, arcosh x, artanh x; Be careful not to confuse these, especially when looking them up in the formulae booklet

反双曲函数是双曲函数的反函数。 记为（arsinh、arcosh、artanh等等）。 与 反三角函数 不同之处是它的前缀是ar意即area (面积)，而不是arc (弧)。 因为双曲角是以双曲线、通过原点直线以及其对x轴的映射三者之间所夹面积定义的，而圆角是以弧长与半径的比值定义。 [1] 我们知道， 三角函数 分为sin（正弦）、cos（余弦）、tan（正切）、cot（余切）、sec（正割）、csc（余割）六种。 而 双曲函数 也如此。 故而，反双曲函数也有六种。 有反双曲正弦、反双曲余弦、反双曲 …

2025年1月31日 · The inverse hyperbolic tangent (Zwillinger 1995, p. 481; Beyer 1987, p. 181), sometimes called the area hyperbolic tangent (Harris and Stocker 1998, p. 267), is the multivalued function that is the inverse function of the hyperbolic tangent. The function is sometimes denoted (Jeffrey 2000, p. 124) or (Gradshteyn and Ryzhik 2000, p. xxx).

2024年8月13日 · They are used to solve equations involving hyperbolic functions and are defined in terms of natural logarithms. How do you find the Inverse Hyperbolic Sine of a Number? I nverse hyperbolic sine, denoted as sinh⁡-1 (x) or arsinh(x), is calculated using the formula: sinh-1 (x) = ln(x + √{x 2 + 1}) What is the Formula for the Inverse ...

Use the logarithmic form of artanh( ) to find the solution in exact form.

If you sub in for the exponential form of tanhy in the LHS and go through the motions, you'll get y (i.e. arctanhx) in terms of x. Then multiply the e^2x+1 across, and subtract the e^2x-1 so you have everything on one side. multiply out the brackets, and try and find new factors composed of y. these can be brought across to create (y+1)/ (y-1).

2018年9月19日 · Prove formula $\operatorname{arctanh} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$ Attempt to solve To start off with definition of functions $\sinh(x)$ and $\cosh(x)$

Use the fact that α and β are both of the form A ± Bi, in conjunction with the fact that i ⋅ lnU − Vi U + Vi = arctanUV 2 , where U = x − A, and V = B. Assuming that you want to do your integrals in terms of real numbers, . . . if b2 − 4ac is zero then neither formula is quite right.