How to get (Taylor) series formula for arcosh?

In summary, the conversation discusses the series for arsinh and attempts to derive it using derivatives and Taylor series. It also mentions a similar series for arcosh and the relationship between the two functions. The conversation concludes with a suggested identity between arsinh and arcosh.
  • #1
swampwiz
571
83
I'm looking at the series published @ Wikipedia: http://en.wikipedia.org/wiki/Inverse_hyperbolic_function

There is a series for arsinh, which I was able to derive with no problem - basically take the derivative of arsinh, which is a radical, then apply the general binomial expansion, which gives a series, from which successive derivatives can be determined, then do a Taylor series @ 0, which for every [ ( f(n) / n! ) xn ] term cancels out every term except the one that has the x power as 0, leaving only one term for each Taylor series term, resulting in the nice ordered series for arsinh. However, I can't seem to get something that appears to go in the direction of the series as presented @ Wikipedia.

It appears that somehow if ( 1 / x ) is used instead of ( x ), then there is some type of series that is similar to that of arsinh (i.e., in the way that the series for cos & sin are related) with a strange ln ( 2 x ) term as well. I figure that there must be some underlying identity between arsinh & arcosh, that somehow related arsinh( x ) with arcosh( 1 / x ) and ln ( 2 x ) but I can't find it anywhere.
 
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  • #2
You can use ##{\displaystyle \operatorname {arsinh} (x)=\operatorname {sgn} (x)\cdot \operatorname {arcosh} \left({\sqrt{x^{2}+1}}\right)}## and for ##{\displaystyle x\geq 1}## we have ##{\displaystyle \operatorname {arcosh} (x)=\operatorname {arsinh} \left({\sqrt {x^{2}-1}}\right)}##.
 

1. How do I find the Taylor series formula for arcosh?

The Taylor series formula for arcosh can be found by taking the derivative of the arcosh function and plugging in the values for x. This will give you the coefficients for the Taylor series expansion.

2. What is the general formula for Taylor series?

The general formula for Taylor series is f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... where f(a) is the value of the function at a and f'(a), f''(a), f'''(a) are the first, second, and third derivatives of the function at a, respectively.

3. Can the Taylor series formula for arcosh be used to approximate other functions?

Yes, the Taylor series formula for arcosh can be used to approximate other functions by finding the coefficients for the Taylor series expansion and plugging in the desired value for x.

4. How many terms should I include in the Taylor series for a good approximation?

The number of terms to include in the Taylor series for a good approximation depends on the desired level of accuracy. Generally, the more terms included, the more accurate the approximation will be.

5. How do I know if the Taylor series for arcosh is convergent?

The Taylor series for arcosh is convergent within a certain interval, which can be determined by using the ratio test or the root test. If the limit of the ratio or root of the terms is less than 1, then the series is convergent within that interval.

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