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Math Help - Principle Value of Cosh

  1. #1
    Super Member craig's Avatar
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    Principle Value of Cosh

    The value of arccosh is given as
    \ln {\color{red}(} x\pm\sqrt{x^2-1}{\color{red})} ,

    but my book states that

    \ln {\color{red}(} x + \sqrt{x^2-1} {\color{red})}

    gives you the principle value. Is the principle value the positive one, the one we would normally take? I'm guessing you get this option due to the graph of \cosh{x} being symmetrical about the y axis?

    Thanks for clearing this one up.
    Last edited by mr fantastic; June 19th 2009 at 06:43 PM. Reason: Fixed typo (added red brackets)
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    Super Member Showcase_22's Avatar
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    I think it has sometihng to do with the graph of cosh always being positive.

    Therefore only ln(x+\sqrt{x^2-1}) is the only one which is always positive.
    Last edited by mr fantastic; June 19th 2009 at 06:41 PM. Reason: Fixed a typo
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  3. #3
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    Quote Originally Posted by Showcase_22 View Post
    I think it has sometihng to do with the graph of cosh always being positive.

    Therefore only ln(x+\sqrt{x^2-1}) is the only one which is always positive.
    A justification of this is required. A justification can be found here: http://www.math.cornell.edu/~kbrown/122/coshinv.pdf
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  4. #4
    Super Member craig's Avatar
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    Thanks, that worksheet explained it really well.
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