Integrate a Hermite_e series over axis 1 in Python

To integrate a Hermite_e series over a specific axis, use the hermite_e.hermeint() method in Python. This function integrates Hermite_e polynomial coefficients and can work with multidimensional arrays where different axes correspond to different variables.

Syntax

hermite_e.hermeint(c, m=1, k=[], lbnd=0, scl=1, axis=0)

Parameters

The function accepts the following parameters:

• c − Array of Hermite_e series coefficients
• m − Order of integration (default: 1)
• k − Integration constant(s) (default: [])
• lbnd − Lower bound of the integral (default: 0)
• scl − Scalar multiplier (default: 1)
• axis − Axis over which integration is performed

Example

Let's create a multidimensional array and integrate over axis 1 −

import numpy as np
from numpy.polynomial import hermite_e as H

# Create a multidimensional array of coefficients
c = np.arange(4).reshape(2,2)

# Display the array
print("Our Array...\n",c)

# Check the Dimensions
print("\nDimensions of our Array...\n",c.ndim)

# Get the Datatype
print("\nDatatype of our Array object...\n",c.dtype)

# Get the Shape
print("\nShape of our Array object...\n",c.shape)

# To integrate a Hermite_e series, use the hermite_e.hermeint() method in Python
print("\nResult...\n",H.hermeint(c, axis = 1))

Our Array...
 [[0 1]
 [2 3]]

Dimensions of our Array...
2

Datatype of our Array object...
int64

Shape of our Array object...
(2, 2)

Result...
 [[0.5 0.  0.5]
 [1.5 2.  1.5]]

How It Works

When integrating over axis 1, the function processes each row independently. For a polynomial represented by coefficients [a, b], the integral becomes [a/2, 0, b/2]. The middle coefficient is 0 because integrating introduces a new term, and the original coefficients are divided by their respective powers.

Conclusion

The hermite_e.hermeint() method efficiently integrates Hermite_e series over specified axes. When using axis=1, integration occurs along rows, producing expanded coefficient arrays with the integrated polynomial representation.