Integrate a Hermite_e series in Python

The Hermite_e series integration can be performed using NumPy's hermite_e.hermeint() method. This method integrates a Hermite_e polynomial series and returns the coefficients of the integrated series.

Syntax

The hermite_e.hermeint() method accepts several parameters ?

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

Parameters

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

Basic Integration Example

Let's integrate a simple Hermite_e series with coefficients [1, 2, 3] ?

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

# Create an array of coefficients
c = np.array([1, 2, 3])

# Display the original coefficients
print("Original coefficients:", c)

# Integrate the Hermite_e series
result = H.hermeint(c)
print("Integrated coefficients:", result)

Original coefficients: [1 2 3]
Integrated coefficients: [1. 1. 1. 1.]

Integration with Custom Parameters

You can specify integration order and constants ?

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

# Coefficients for Hermite_e series
coefficients = np.array([2, 4, 6])

# Integration with order m=1 and integration constant k=5
result1 = H.hermeint(coefficients, m=1, k=5)
print("Integration with k=5:", result1)

# Integration with order m=2
result2 = H.hermeint(coefficients, m=2)
print("Double integration:", result2)

Integration with k=5: [5. 2. 2. 2.]
Double integration: [0. 0. 1. 1. 1.]

How It Works

The integration increases the degree of the polynomial by 1 for each integration order. The original polynomial c? + c?x + c?x² becomes a polynomial of higher degree after integration, with the integration constant added as the first coefficient.

Conclusion

Use hermite_e.hermeint() to integrate Hermite_e polynomial series in NumPy. The method returns coefficients of the integrated polynomial, with customizable integration order and constants.