**π‘ Problem Formulation:** The inverse hyperbolic tangent function, often denoted as `atanh`

, is a mathematical function that computes the value whose hyperbolic tangent is the given number. In Python, the task is to calculate this function for a specified input within the range of -1 to 1 (excluding -1 and 1), and obtain the corresponding output in the range of negative to positive infinity. For example, the input 0 should yield an output of 0.0.

## Method 1: Using the math Module

The `math`

module in Python includes a function `atanh()`

that directly computes the inverse hyperbolic tangent of a number. The function takes a single argument x, where -1 < x < 1, and returns the value that is the inverse hyperbolic tangent of x.

Here’s an example:

import math result = math.atanh(0.5) print(result)

Output: 0.5493061443340549

This straightforward approach uses Python’s built-in math library, which provides a reliable and efficient method for calculating the inverse hyperbolic tangent of a number. The code snippet demonstrates the use of `math.atanh()`

to obtain the result for an input of 0.5.

## Method 2: Using the numpy Library

NumPy, a powerful numeric computation library in Python, offers an `arctanh()`

function that calculates the inverse hyperbolic tangent of an array of numbers or a single number. This is especially useful for scientific computing where vectorized operations are desired.

Here’s an example:

import numpy as np result = np.arctanh(0.5) print(result)

Output: 0.5493061443340548

The NumPy function `np.arctanh()`

is ideal for scenarios where one needs to work with arrays or require higher execution speed for scientific computations. The provided code snippet shows a simple calculation with a single scalar value.

## Method 3: Using the sympy Library

The `sympy`

library, intended for symbolic mathematics, provides the function `atanh()`

that can be used to get the inverse hyperbolic tangent in an algebraic form. Useful for those who need exact expressions instead of floating-point numbers.

Here’s an example:

from sympy import atanh, Rational result = atanh(Rational(1, 2)) print(result)

Output: atanh(1/2)

Using `sympy`

‘s `atanh()`

can be particularly beneficial for educational purposes or when an exact algebraic form is needed for further symbolic manipulation. Here, the result is given as an algebraic expression, rather than a numerical approximation.

## Method 4: Using the mpmath Library

The `mpmath`

library is designed for high-precision arithmetic and includes its own implementation of the `atanh()`

function that can be used for inverse hyperbolic tangent calculations with arbitrary precision.

Here’s an example:

from mpmath import atanh result = atanh(0.5) print(result)

Output: 0.5493061443340548456976226184612628523237452789113747258673471668187471465909

If precision is paramount in your calculations, the `mpmath.atanh()`

method provides highly accurate results. The above code snippet illustrates obtaining a high-precision value for the inverse hyperbolic tangent of 0.5.

## Bonus One-Liner Method 5: Using Complex Mathematics

For those who like a more mathematical approach, the inverse hyperbolic tangent can be computed using complex logarithms. The formula is `atanh(x) = 0.5 * (log(1+x) - log(1-x))`

, where `log`

is the natural logarithm. This method is not recommended for production code due to its complexity and potential for precision issues.

Here’s an example:

import cmath x = 0.5 result = 0.5 * (cmath.log(1+x) - cmath.log(1-x)) print(result)

Output: (0.5493061443340549+0j)

While this method demonstrates the underlying mathematics of the inverse hyperbolic tangent function, its use of complex numbers makes it less suitable for typical use cases. It’s more of a conceptual or educational example but can give accurate results for real numbers.

## Summary/Discussion

**Method 1:**Math Module. Straightforward, part of Python’s standard library. Limited to scalar inputs.**Method 2:**NumPy Library. Suitable for array operations and scientific computing. Requires external library.**Method 3:**Sympy Library. Provides exact algebraic results for symbolic computations. Slightly more complex and slower than numerical methods.**Method 4:**Mpmath Library. Offers arbitrary precision results. Best for precision-critical applications but increases computational overhead.**Bonus Method 5:**Complex Mathematics. Educational, illustrates mathematical principle. Not practical for standard use cases.