Deriving the Taylor Series for the Inverse Hyperbolic Cosine Function

The inverse hyperbolic cosine function, denoted as cosh^{-1}x, is a significant function in both theory and application. This article will guide you through the process of deriving the Taylor series for cosh^{-1}x around the point x 1. The Taylor series is a powerful tool for approximating functions and is often used in numerical methods and mathematical analysis.

The inverse hyperbolic cosine function, defined for x geq; 1, is given by:

[ cosh^{-1}x ln x sqrt{x^2 - 1} ]

Step 1: Finding the Derivatives

To construct the Taylor series for cosh^{-1}x, we need to find its derivatives at a specific point. A common choice is x 1. Here's how we proceed:

Zeroth derivative: [ cosh^{-1}1 0 ] First derivative: Using the chain rule: [ frac{d}{dx} cosh^{-1}x frac{1}{sqrt{x^2 - 1}} ]

Evaluating at x 1 gives:

[ frac{d}{dx} cosh^{-1}xvert_{x1} frac{1}{sqrt{1^2 - 1}} text{undefined} ]

Since the first derivative is undefined at x 1, we need to find higher derivatives or consider a series expansion.

Step 2: Series Expansion

To find the Taylor series around x 1, we use the fact that cosh^{-1}x can be expanded using the binomial series for sqrt{x^2 - 1}:

Solving for a simpler form, let (x 1 u), where (u) is small:

[ cosh^{-1}(1 u) ln (1 u) sqrt{(1 u)^2 - 1} ln (1 u) sqrt{u^2 2u} ]

Expanding (sqrt{u^2 2u}) using the binomial expansion for small (u):

[ sqrt{u^2 2u} approx sqrt{2u} sqrt{2} sqrt{u} quad text{for small } u ]

Putting it together, we have:

[ cosh^{-1}(1 u) approx ln(1 u) sqrt{2u} ]

The Taylor series expansion for (ln(1 u)) and (sqrt{2u}) around (u 0) is:

[ ln(1 u) u - frac{u^2}{2} frac{u^3}{3} - frac{u^4}{4} cdots ] [ sqrt{2u} sqrt{2} sqrt{u} sqrt{2} sqrt{u} ]

Multiplying these expansions, we get:

[ cosh^{-1}(1 u) approx (1 2u) left(frac{u}{1} - frac{u^2}{2} frac{u^3}{3} - frac{u^4}{4} cdots right) ]

This leads to the series:

[ cosh^{-1}(1 u) approx u - frac{u^2}{4} frac{u^4}{12} - frac{u^4}{32} cdots ]

Step 3: Taylor Series Representation

Putting everything together, the Taylor series for (cosh^{-1}x) expanded around (x 1) is:

[ cosh^{-1}x x - 1 - frac{(x - 1)^2}{4} frac{(x - 1)^3}{12} - frac{(x - 1)^4}{32} cdots ]

This series converges for (x) in the interval ([1, infty]).

Final Representation:

The Taylor series for (cosh^{-1}x) around (x 1) is:

[ cosh^{-1}x sum_{n1}^{infty} frac{(x - 1)^n}{2^{n-1} n!} ]

This series converges for (x) in the interval ([1, infty]).

Conclusion

The Taylor series for the inverse hyperbolic cosine function around (x 1) provides a powerful approximation method. This series is particularly useful in numerical analysis and solving equations involving hyperbolic functions. By understanding and applying the steps outlined, you can effectively extend this knowledge to other functions and mathematical problems.