Hyperbolic Functions in NumPy

Hyperbolic functions are mathematical functions that are analogous to the familiar trigonometric functions, but are based on hyperbolas instead of circles. They play a significant role in various fields, including solving differential equations, modeling physical phenomena, and analyzing systems exhibiting exponential growth or decay.

NumPy provides an efficient and vectorized suite of functions for computing both standard and inverse hyperbolic functions, making them readily accessible for array operations in scientific computing.

Standard Hyperbolic Functions

These functions compute the hyperbolic sine, cosine, and tangent of an input array.

numpy.sinh(x)

The hyperbolic sine of x is mathematically defined as:

$ \text{sinh}(x) = \frac{e^x - e^{-x}}{2} $

This function calculates the hyperbolic sine for each element in the input array x.

Example:

import numpy as np

values = np.array([0, 1, 2, 3])
sinh_values = np.sinh(values)
print("Hyperbolic Sine values:", sinh_values)

Output:

Hyperbolic Sine values: [ 0.          1.17520119  3.62686041 10.01787493]

numpy.cosh(x)

The hyperbolic cosine of x is defined as:

$ \text{cosh}(x) = \frac{e^x + e^{-x}}{2} $

This function computes the hyperbolic cosine for each element in the input array x.

Example:

import numpy as np

values = np.array([0, 1, 2, 3])
cosh_values = np.cosh(values)
print("Hyperbolic Cosine values:", cosh_values)

Output:

Hyperbolic Cosine values: [ 1.          1.54308063  3.76219569 10.067662  ]

numpy.tanh(x)

The hyperbolic tangent of x is defined as the ratio of hyperbolic sine to hyperbolic cosine:

$ \text{tanh}(x) = \frac{\text{sinh}(x)}{\text{cosh}(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} $

This function calculates the hyperbolic tangent for each element in the input array x.

Example:

import numpy as np

values = np.array([0, 1, 2, 3])
tanh_values = np.tanh(values)
print("Hyperbolic Tangent values:", tanh_values)

Output:

Hyperbolic Tangent values: [0.         0.76159416 0.96402758 0.99505475]

Inverse Hyperbolic Functions

These functions compute the inverse hyperbolic sine, cosine, and tangent of an input array.

numpy.arcsinh(x)

The inverse hyperbolic sine (also known as the area hyperbolic sine) is defined as:

$ \text{arcsinh}(x) = \log(x + \sqrt{x^2 + 1}) $

This function computes the inverse hyperbolic sine for each element in the input array x.

Example:

import numpy as np

values = np.array([0, 1, 2, 3])
asinh_values = np.arcsinh(values)
print("Inverse Hyperbolic Sine values:", asinh_values)

Output:

Inverse Hyperbolic Sine values: [0.         0.88137359 1.44363548 1.81844646]

numpy.arccosh(x)

The inverse hyperbolic cosine (also known as the area hyperbolic cosine) is defined as:

$ \text{arccosh}(x) = \log(x + \sqrt{x^2 - 1}) $

Note: This function is only defined for real values when $ x \ge 1 $. For inputs less than 1, it will return NaN.

Example:

import numpy as np

values = np.array([1, 2, 3, 4])
acosh_values = np.arccosh(values)
print("Inverse Hyperbolic Cosine values:", acosh_values)

Output:

Inverse Hyperbolic Cosine values: [0.         1.3169579  1.76274717 2.06343707]

numpy.arctanh(x)

The inverse hyperbolic tangent (also known as the area hyperbolic tangent) is defined as:

$ \text{arctanh}(x) = \frac{1}{2} \log\left(\frac{1 + x}{1 - x}\right) $

Note: This function is only defined for real values when $ -1 < x < 1 $. For inputs outside this range, it will return NaN.

Example:

import numpy as np

values = np.array([0, 0.5, 0.9])
atanh_values = np.arctanh(values)
print("Inverse Hyperbolic Tangent values:", atanh_values)

Output:

Inverse Hyperbolic Tangent values: [0.         0.54930614 1.47221949]

Conclusion

NumPy's hyperbolic functions offer a computationally efficient and convenient way to perform mathematical operations involving hyperbolic and inverse hyperbolic values. These functions are invaluable in diverse scientific and engineering disciplines, particularly when modeling phenomena characterized by exponential growth, decay, or wave-like behavior.

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