NumPy - Hyperbolic Functions

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NumPy Hyperbolic Functions

The hyperbolic functions are similar to trigonometric functions but are based on hyperbolas instead of circles. The most common hyperbolic functions are sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent).

In NumPy, there are several hyperbolic functions available to calculate these values for arrays of numbers. These functions are widely used in scenarios involving hyperbolic curves or exponential growth and decay processes.

Hyperbolic Sine Using sinh() Function

The numpy.sinh() function calculates the hyperbolic sine of each element in the input array. The hyperbolic sine is defined as −

sinh(x) = (ex - e(-x)) / 2

Example: Hyperbolic Sine

In the following example, we calculate the hyperbolic sine of an array of values using NumPy's sinh() function −

import numpy as np

# Define an array of values
values = np.array([0, 1, 2, 3])

# Calculate the hyperbolic sine of each element
sinh_values = np.sinh(values)

print("Hyperbolic Sine values:", sinh_values)

Hyperbolic Sine values: [ 0.          1.17520119  3.62686041 10.01787493]

Hyperbolic Cosine Using cosh() Function

The numpy.cosh() function calculates the hyperbolic cosine of each element in the input array. The hyperbolic cosine is defined as −

cosh(x) = (ex + e(-x)) / 2

Example: Hyperbolic Cosine

In the following example, we calculate the hyperbolic cosine of an array of values using NumPy's cosh() function −

import numpy as np

# Define an array of values
values = np.array([0, 1, 2, 3])

# Calculate the hyperbolic cosine of each element
cosh_values = np.cosh(values)

print("Hyperbolic Cosine values:", cosh_values)

This will produce the following result −

Hyperbolic Cosine values: [ 1.          1.54308063  3.76219569 10.067662  ]

Hyperbolic Tangent Using tanh() Function

The numpy.tanh() function calculates the hyperbolic tangent of each element in the input array. The hyperbolic tangent is defined as −

tanh(x) = sinh(x) / cosh(x)

Example: Hyperbolic Tangent

In the following example, we calculate the hyperbolic tangent of an array of values using NumPy's tanh() function −

import numpy as np

# Define an array of values
values = np.array([0, 1, 2, 3])

# Calculate the hyperbolic tangent of each element
tanh_values = np.tanh(values)

print("Hyperbolic Tangent values:", tanh_values)

Following is the output of the above code −

Hyperbolic Tangent values: [0.         0.76159416 0.96402758 0.99505475]

Inverse Hyperbolic Sine

The numpy.arcsinh() function calculates the inverse hyperbolic sine of each element in the input array. The inverse hyperbolic sine is defined as −

arcsinh(x) = log(x + sqrt(x2 + 1))

Example: Inverse Hyperbolic Sine

In the following example, we calculate the inverse hyperbolic sine of an array of values using NumPy's arcsinh() function −

import numpy as np

# Define an array of values
values = np.array([0, 1, 2, 3])

# Calculate the inverse hyperbolic sine of each element
asinh_values = np.arcsinh(values)

print("Inverse Hyperbolic Sine values:", asinh_values)

The output obtained is as shown below −

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

Inverse Hyperbolic Cosine

The numpy.arccosh() function calculates the inverse hyperbolic cosine of each element in the input array. The inverse hyperbolic cosine is defined as −

arccosh(x) = log(x + sqrt(x2 - 1))

Example: Inverse Hyperbolic Cosine

In the following example, we calculate the inverse hyperbolic cosine of an array of values using NumPy's arccosh() function −

import numpy as np

# Define an array of values
values = np.array([1, 2, 3, 4])

# Calculate the inverse hyperbolic cosine of each element
acosh_values = np.arccosh(values)

print("Inverse Hyperbolic Cosine values:", acosh_values)

After executing the above code, we get the following output −

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

Inverse Hyperbolic Tangent

The numpy.arctanh() function calculates the inverse hyperbolic tangent of each element in the input array. The inverse hyperbolic tangent is defined as −

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

Example: Inverse Hyperbolic Tangent

In the following example, we calculate the inverse hyperbolic tangent of an array of values using NumPy's arctanh() function −

import numpy as np

# Define an array of values
values = np.array([0, 0.5, 0.9])

# Calculate the inverse hyperbolic tangent of each element
atanh_values = np.arctanh(values)

print("Inverse Hyperbolic Tangent values:", atanh_values)

The result produced is as follows −

Inverse Hyperbolic Tangent values: [0.         0.54930614 1.47221949]