Hyperbolic functions play a crucial role in advanced mathematics, physics, and engineering. Just as trigonometric functions describe circles, hyperbolic functions describe hyperbolas. NumPy provides fast, vectorized universal functions (ufuncs) to compute these functions efficiently.

In this article, you'll learn how to use NumPy's hyperbolic ufuncs, including their inverses, use cases, and best practices.

What Are Hyperbolic Functions?

Hyperbolic functions are analogs of trigonometric functions but for a hyperbola. The primary hyperbolic functions include:

They are useful in:

• Relativity and spacetime geometry

• Signal processing

• Calculus and differential equations

• Neural network activations (e.g. tanh)

NumPy Hyperbolic ufuncs

✅ Basic Hyperbolic Functions

Inverse Hyperbolic Functions

Step-by-Step Code Examples

1. Basic Hyperbolic Functions

import numpy as np

x = np.array([-2, -1, 0, 1, 2])

print("sinh(x):", np.sinh(x))
print("cosh(x):", np.cosh(x))
print("tanh(x):", np.tanh(x))

✅ Output:

sinh(x): [-3.62686 -1.1752  0.       1.1752  3.62686]
cosh(x): [3.7622  1.5431  1.      1.5431  3.7622]
tanh(x): [-0.9640 -0.7615  0.      0.7615  0.9640]

2. Inverse Hyperbolic Functions

x = np.array([-2, -1, 0, 1, 2])

print("arcsinh(x):", np.arcsinh(x))   # Defined for all real x
print("arccosh(x):", np.arccosh(x[x >= 1]))  # Defined for x >= 1
print("arctanh(x):", np.arctanh(x[np.abs(x) < 1]))  # |x| < 1

✅ Output:

arcsinh(x): [-1.4436 -0.8814  0.      0.8814  1.4436]
arccosh(x): [0.      1.3169]
arctanh(x): [-0.5493  0.      0.5493]

3. Graphing Hyperbolic Functions (Optional)

import matplotlib.pyplot as plt

x = np.linspace(-4, 4, 400)
plt.plot(x, np.sinh(x), label='sinh(x)')
plt.plot(x, np.cosh(x), label='cosh(x)')
plt.plot(x, np.tanh(x), label='tanh(x)')
plt.title("Hyperbolic Functions")
plt.xlabel("x")
plt.ylabel("y")
plt.grid(True)
plt.legend()
plt.show()

✅ Full Working Example

import numpy as np

# Input array
x = np.linspace(-2, 2, 5)

print("Input:", x)

# Basic hyperbolic functions
print("sinh:", np.sinh(x))
print("cosh:", np.cosh(x))
print("tanh:", np.tanh(x))

# Inverse functions
print("arcsinh:", np.arcsinh(x))
print("arccosh (x >= 1):", np.arccosh(x[x >= 1]))
print("arctanh (|x| < 1):", np.arctanh(x[np.abs(x) < 1]))

⚠️ Common Pitfalls

Tips and Best Practices

• Use np.clip(x, -0.9999, 0.9999) to avoid arctanh domain errors.

• Combine tanh with np.gradient for signal smoothing or activation functions.

• For large inputs, consider logarithmic equivalents of inverse hyperbolic functions to avoid overflow.

• Plot functions to visualize growth or decay trends.

Mathematical Reference

Summary Table

Conclusion

NumPy's hyperbolic ufuncs provide a fast, reliable, and vectorized way to handle computations involving hyperbolic geometry, signal modeling, and advanced mathematics. With careful domain handling and input validation, these tools can be seamlessly integrated into a wide range of scientific computing applications.