# 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?

Thanks.

## 14 Replies to “MitPY 12: Inverse Hyperbolic Functions”

1. John Friend says:

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:

.

1. Craig says:

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. John Friend says:

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

1. Craig says:

3. 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. John Friend says:

Indeed. The range of is and so the domain of is .

2. Craig says:

Do you still need to give a restriction if x is complex?

1. John Friend says:

has no solution therefore …?

1. John Friend says:

But since the complex logarithm is multi-valued, you need to use the Principal Value log …

2. Craig says:

… there still needs to be a restriction.

1. John Friend says:

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.

4. Glen says:

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

1. Craig says:

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.

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