This MitPY comes from occasional commenter, Craig:
Hello folks, just a quick (probably stupid) question. Is the following step justified?
Thanks.
This MitPY comes from occasional commenter, Craig:
Hello folks, just a quick (probably stupid) question. Is the following step justified?
Thanks.
Hi Craig. I’d say yes (but I can think of easier ways (for me, anyway) of getting
than deriving(?) the first identity).
Here is how I would justify it:
Let
be defined from the identity 
.
To get the inverse, let
where 
.
Then from the above identity:
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.
Hi Craig.
There’s a small glitch at the moment which means I can’t see your post (but I got an email alert of your post). I’ve attached part of my notes I give my students – I hope they help.
Inverse hyperbolic functions
Typo in attachment. Correct copy attached.
Inverse hyperbolic functions
Your notes did help, thanks!
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).
Indeed. The range of
is 
and so the domain of 
is 
.
Do you still need to give a restriction if x is complex?
But since the complex logarithm is multi-valued, you need to use the Principal Value log …
… there still needs to be a restriction.
There are no restrictions on
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.
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
.
But is that variable
the *same* 
as in the second equation? Or is it something different?
Cheers
Glen
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.