MitPY 12: Inverse Hyperbolic Functions

This MitPY comes from occasional commenter, Craig:

Hello folks, just a quick (probably stupid) question. Is the following step justified?

\boldsymbol{x = \ln\left(\cosh x + \sqrt{(\cosh x)^2-1}\right)}

\boldsymbol{\mbox{\bf arccosh}\, x = \ln\left(x+\sqrt{x^2 -1}\right)}

Thanks.

14 Replies to “MitPY 12: Inverse Hyperbolic Functions”

  1. Hi Craig. I’d say yes (but I can think of easier ways (for me, anyway) of getting \text{arccosh} (x) than deriving(?) the first identity).

    Here is how I would justify it:

    Let y = \cosh (x) be defined from the identity \displaystyle x = \ln \left( y + \sqrt{y^2 - 1} \right).

    To get the inverse, let \displaystyle x = \cosh (y) where y = \text{arccosh} (x).

    Then from the above identity:

    \displaystyle y = \ln \left( x + \sqrt{x^2 - 1} \right).

    1. Thanks John. I had to show that arccosh was actually ln(stuff) in disguise, so I tried starting with the exponential definition for cosh then try and get arccosh somehow. I got the right formula but x and arccoshx were in each others place. Just wondered if I could swap them around. And I still have to explain how I got rid of a plus/minus at some point. But I’m happy to know that the last two lines are fine as you say.

  2. arccosh(x) is only defined for x ≥ 1, so you have to add that in as a condition as well.

    Once you do that, yes, this is fine and justified, since if f(x) = g(x), then f(h(x)) = g(h(x)) (assuming everything in question exists).

          1. There are no restrictions on z but since the complex logarithm is multi-valued you need to use the Principal Value log and so there are restrictions on the values of the complex logarithm.

  3. Hi Craig!

    I think it is always worth thinking about if you are consistent with notation and variables. In the first equation, I guess you know how to argue that it makes sense for all real x.

    But is that variable x the *same* x as in the second equation? Or is it something different?

    Cheers
    Glen

    1. Thanks Glen. In the first equation, x means the input for the cosh function. For the second equation, x means the input for the inverse cosh function. So it looks like they are different xs. I just had to get to the second equation and I got to the first so I thought I would take a shortcut and see if that shortcut made sense and that’s why I’m here. So I guess I should have changed the x in the second equation to something else. Or actually the first x to something else. Which is I think what happens in John’s notes.

Leave a Reply

Your email address will not be published. Required fields are marked *

The maximum upload file size: 128 MB. You can upload: image, audio, video, document, spreadsheet, interactive, text, archive, code, other. Links to YouTube, Facebook, Twitter and other services inserted in the comment text will be automatically embedded. Drop file here

This site uses Akismet to reduce spam. Learn how your comment data is processed.