NumPy arctanh() Function

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The NumPy arctanh() function computes the inverse hyperbolic tangent (area hyperbolic tangent) of each element in the input array. The inverse hyperbolic tangent is also known as atanh or tanh^-1 and is mathematically defined as −

arctanh(x) = 0.5 * ln((1 + x) / (1 - x))

The function is the inverse of the hyperbolic tangent function (tanh()) and returns values in radians. The arctanh() is a multivalued function, for each real number x there are infinitely many numbers of complex number z such that tanh(z) = x. The convention is to return the z whose imaginary part lies in [-pi/2, pi/2].

For real-valued input data types, arctanh always returns real output. For each value that cannot be expressed as a real number or infinity, it yields nan and sets the invalid floating point error flag.

For complex-valued input, arctanh() is a complex analytical function that ranges between [-1, -infinite] and [1, infinite] and is continuous from above on the former and from below on the latter.

Syntax

Following is the syntax of the NumPy arctanh() function −

numpy.arctanh(x, out=None, where=True, casting='same_kind', order='K', dtype=None, subok=True)

Parameters

Following are the parameters of the NumPy arctanh() function −

• x: Input array, it can be a NumPy array, list, or scalar value. The elements must satisfy -1 < x < 1 for real results.
• out (optional): Alternate output array to place the result. It must have the same shape as the expected output.
• where (optional): A Boolean array. If True, compute the result; otherwise, it leaves the corresponding output elements unchanged.
• dtype (optional): Specifies the data type of the result.
• casting (optional): Ensures equivalent type conversion occurs. For example, converting from `float32` to `float64` is allowed, but converting from `float64` to `int32` is not.
• subok (optional): Determines whether to subclass the output array if the data type is changed or to return a base-class array.
• order (optional): Specifies the memory layout of the array −
• 'C': C-style row-major order.
• 'F': Fortran-style column-major order.
• 'A': 'F' if the input is Fortran contiguous, 'C' otherwise.
• 'K': This is the default value. Keeps the order as close as possible to the input.

Return Values

This function returns a NumPy array with the inverse hyperbolic tangent of the input values. If an input value is outside the range it results nan.

Example

Following is a basic example to compute the inverse hyperbolic tangent of each element in an array using the NumPy arctanh() function −

import numpy as np
# input array
x = np.array([-0.5, 0, 0.5])
# applying arctanh
result = np.arctanh(x)
print("Arctanh Result:", result)

Output

Following is the output of the above code −

Arctanh Result: [-0.54930614  0.          0.54930614]

Example: Scalar Input as an Argument

The arctanh() function also accepts a scalar input. In the following example, we have passed 0.5 as an argument to the arctanh() function −

import numpy as np
# scalar input
scalar = 0.5
# applying arctanh
result = np.arctanh(scalar)
print("Arctanh Result for Scalar Input:", result)

Output

Following is the output of the above code −

Arctanh Result for Scalar Input: 0.5493061443340548

Example: Multi-dimensional Array

The arctanh() function operates on multi-dimensional arrays. In the following example, we have created a 2X2 NumPy array, where each element computes its inverse hyperbolic tangent −

import numpy as np
# 2D array
x = np.array([[-0.5, 0], [0.5, 0.8]])
result = np.arctanh(x)
print("Arctanh Result for 2D Array:\n", result)

Output

Following is the output of the above code −

Arctanh Result for 2D Array:
 [[-0.54930614  0.        ]
  [ 0.54930614  1.09861229]]

Example: Plotting 'arctanh()' Function

In the following example, we have plotted the behaviour of the arctanh() function. To achieve this, we need to import the numpy and matplotlib.pyplot modules −

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-0.99, 0.99, 100)  # input range (-1 < x < 1)
y = np.arctanh(x)  # applying arctanh
plt.plot(x, y)
plt.title("Inverse Hyperbolic Tangent Function")
plt.xlabel("Input")
plt.ylabel("arctanh(x)")
plt.grid()
plt.show()