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There are several results in Spiegel's "Mathematical Handbook of Formulas and Tables" (Schaum, 1968) concerning $\cosh^{-1}$ which are presented in the following format:

$$\cosh^{-1} x = \pm \left\{{\ln (2x) - \left({\dfrac 1 {2 \cdot 2x^2} + \dfrac {1 \cdot 3} {2 \cdot 4 \cdot 4x^4} + \dfrac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6 \cdot 6x^6} }\right) }\right\}$$

$$[+ \text { if }\cosh^{-1} x > 0, x \ge 1, - \text { if }\cosh^{-1} x < 0, x \ge 1]$$

(item $20.40$)

and:

$$\int \cosh^{-1} \frac x a \, \mathrm d x = \begin{cases} x \cosh^{-1} (x/a) - \sqrt {x^2 - a^2} & : \cosh^{-1} (x/a) > 0 \\ x \cosh^{-1} (x/a) + \sqrt {x^2 - a^2} & : \cosh^{-1} (x/a) < 0 \end{cases}$$

(item $14.651$)

There are a number of such.

What confuses me is that I have been led to believe that the (real) inverse hyperbolic cosine is specifically defined as being the positive value of $y$ such that $\cosh y = x$.

So what is the motivation and conventional interpretation of "$f (\cosh^{-1} x)$ such that $\cosh^{-1} x < 0$"?

Prime Mover
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2 Answers2

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Assuming $x, y$ real, and $x \ge 1$, since

$$\mathrm{cosh}(y) = \mathrm{cosh}(-y) =x $$

one has 2 possible choices for the definition of a single valued function $\mathrm{cosh}^{-1}(x)$, namely

$$\mathrm{cosh}^{-1}(x) =y$$

or

$$\mathrm{cosh}^{-1}(x) =-y$$

It seems like the author is just covering all the cases, since it's a reference text.

Andy Walls
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The 1978 edition of the Mathematical Handbook says that "$\text{Arg ch}(x)=\ln(x+\sqrt{x^2-1}),x\ge0$, ($\text{Arg ch}(x)>0$ is the principal branch)" (translated from French). This fully justifies the formulas that you mention, which pertain to the two possible branches (the two solutions in $y$ of $x=\cosh(y)$).

Technically speaking, the hyperbolic cosine is not an invertible function, and the notation $\cosh^{-1}(x)$ is no so valid. The function $\text{arcosh}(x)$ is well-known and "by default" taken to be the principal branch.

  • Well okay, but the 1968 edition (the only one I have immediate access to) actually has $\cosh^{-1}$ rather than $\operatorname{arcosh}$ ... so in light of what you say, perhaps Spiegel is insisting that the negative branch is considered as a separate entity. – Prime Mover Jan 02 '21 at 13:16
  • @PrimeMover: the situation seems similar to the existence of two square roots of a positive number, while the square root function has of course a unique value. I claim that $\cosh^{-1}(x)=\pm\cdots$ is a dubious notation. –  Jan 02 '21 at 13:20
  • Thanks for confirming my suspicions, Yves -- I will construct my pages accordingly, along with a note confirming what you said. I think it's important for this to be made explicit. – Prime Mover Jan 02 '21 at 13:40
  • I have taken the time and effort to go through my website and make this important distinction. Many thanks for the pointer. – Prime Mover Jan 09 '21 at 16:57