Integrate a Hermite_e series and set the order of integration in Python

To integrate a Hermite_e series, use the hermite_e.hermeint() method in Python. This function performs polynomial integration on Hermite_e series coefficients with customizable integration order and constants.

Syntax

numpy.polynomial.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 (must be positive, default: 1)
• k - Integration constant(s) (default: [])
• lbnd - Lower bound of the integral (default: 0)
• scl - Scalar multiplier applied after each integration (default: 1)
• axis - Axis over which the integral is taken (default: 0)

Example

Let's integrate a Hermite_e series with different integration orders ?

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 array
print("Coefficients:", c)
print("Shape:", c.shape)

# Integrate with order m = 1 (default)
result1 = H.hermeint(c)
print("\nIntegration order 1:")
print(result1)

# Integrate with order m = 2
result2 = H.hermeint(c, m=2)
print("\nIntegration order 2:")
print(result2)

# Integrate with order m = 3
result3 = H.hermeint(c, m=3)
print("\nIntegration order 3:")
print(result3)

Coefficients: [1 2 3]
Shape: (3,)

Integration order 1:
[0. 1. 1. 1.]

Integration order 2:
[0.   0.   0.5  0.33333333 0.25      ]

Integration order 3:
[ 0.          0.25       -0.25        0.5         0.16666667  0.08333333
  0.05      ]

Integration with Constants

You can specify integration constants using the k parameter ?

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

c = np.array([1, 2, 3])

# Integration with constant k = 5
result_with_constant = H.hermeint(c, m=1, k=[5])
print("With integration constant k=5:")
print(result_with_constant)

# Integration with multiple constants for higher order
result_multi_const = H.hermeint(c, m=2, k=[2, 3])
print("\nWith multiple constants [2, 3]:")
print(result_multi_const)

With integration constant k=5:
[5. 1. 1. 1.]

With multiple constants [2, 3]:
[2.         3.         0.5        0.33333333 0.25      ]

How It Works

The integration process follows these steps:

1. Each coefficient is integrated according to Hermite_e polynomial rules
2. The result is multiplied by the scaling factor scl
3. Integration constants are added based on the lower bound lbnd
4. Higher order integration repeats this process m times

Conclusion

The hermeint() function provides flexible integration of Hermite_e series with customizable order and constants. Higher integration orders increase the polynomial degree and modify the coefficient structure accordingly.