# Why is Mathematica giving me fits when I try to evaluate ArcCosh[Sqrt[2]]?

• AxiomOfChoice
In summary, the conversation discusses the issue of Mathematica and Maple not being able to comply or evaluate the expression \cosh^{-1}(\sqrt{2}). The reason for this is not clear and there are suggestions for alternative expressions and decimal approximations.
AxiomOfChoice
For some reason, Mathematica will not comply when I try to determine

$$\cosh^{-1}(\sqrt{2})$$

Why is this the case? Is $$\cosh^{-1}(x)$$ undefined there or something? If so, why? I don't really see it...

Thanks!

UPDATE: Looks like Maple 11 refuses to evaluate $$\cosh^{-1}(\sqrt{2})$$ too. What is going on here?!

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Don't know, have you tried ln(x+root(x^2-1)) instead?

http://www.research.att.com/~njas/sequences/A091648

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It could be a problem if you have replaced this $\sqrt{2}$ with this $2^{1/2}$.

CRGreathouse said:
http://www.research.att.com/~njas/sequences/A091648
If the OP is looking for a decimal approximation, then something along the lines of Round[ArcCosh[Sqrt[2]],0.0001] should work just as well on Mathematica.

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Gokul43201 said:
If the OP is looking for a decimal approximation, then something along the lines of Round[ArcCosh[Sqrt[2]],0.0001] should work just as well on Mathematica.

I thought the additional information there (inflection point, etc) might be relevant the the unstated underlying problem.

UPDATE: Looks like Maple 11 refuses to evaluate "cosh^{-1}(\\sqrt{2})" too. What is going on here

Hi AxiomOfChoice, what exactly do you mean by evaluate in this case? Maple will try to symbolically simply an expression, it won't return a decimal approximation unless you tell it to (typically using the "evalf" function.

I've got Maple 7 and "evalf(arccosh(sqrt(2)));" returns 0.8813735866.

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## 1. Why is Mathematica giving me an error when I try to evaluate ArcCosh[Sqrt[2]]?

Mathematica is giving you an error because the input expression, ArcCosh[Sqrt[2]], is undefined for real numbers. This is because the argument of ArcCosh must be greater than or equal to 1, and Sqrt[2] is less than 1.

## 2. How can I fix the error when evaluating ArcCosh[Sqrt[2]] in Mathematica?

To fix the error, you can use the complex version of ArcCosh by inputting ArcCosh[Sqrt[2]]//ComplexExpand. This will give you the answer in terms of complex numbers instead of real numbers.

## 3. Why is Mathematica giving me a different result than expected for ArcCosh[Sqrt[2]]?

Mathematica uses the principal branch of the complex logarithm when evaluating functions like ArcCosh. This means that the result may be different from what you expect if you are not familiar with complex numbers and branches of functions.

## 4. Is there a way to get the result I want for ArcCosh[Sqrt[2]] in Mathematica?

Yes, you can specify a different branch of the complex logarithm by using the option Branch -> n in the input expression, where n is an integer representing the desired branch. For example, ArcCosh[Sqrt[2], Branch -> 2] will give you the result you want.

## 5. What is the purpose of the ComplexExpand function in Mathematica?

The ComplexExpand function in Mathematica is used to simplify expressions involving complex numbers. It expands complex functions into their real and imaginary parts, making it easier to work with complex expressions and functions in Mathematica.

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