Integrate a Hermite_e series and set the Integration constant in Python

To integrate a Hermite_e series, use the hermite_e.hermeint() method in Python. This function integrates a Hermite_e polynomial and allows you to set integration constants.

Syntax

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 constant(s). If k == [], all constants are zero (default: [])
• lbnd ? Lower bound of integration (default: 0)
• scl ? Scalar multiplier applied after each integration (default: 1)
• axis ? Axis over which integration is performed (default: 0)

Basic Integration

Let's integrate a simple Hermite_e series with default parameters ?

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

# Create coefficient array
c = np.array([1, 2, 3])
print("Original coefficients:", c)

# Basic integration
result = H.hermeint(c)
print("Integrated series:", result)

Original coefficients: [1 2 3]
Integrated series: [0. 1. 1. 1.]

Setting Integration 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])

# Set integration constant k = 5
result_with_constant = H.hermeint(c, k=5)
print("With k=5:", result_with_constant)

# Multiple integration with different constants
result_double = H.hermeint(c, m=2, k=[2, 3])
print("Double integration with k=[2, 3]:", result_double)

With k=5: [5. 1. 1. 1.]
Double integration with k=[2, 3]: [2. 3. 0.5 0.33333333 0.33333333]

Higher Order Integration

The m parameter controls the order of integration ?

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

c = np.array([2, 4, 6])

# First order integration (default)
first_order = H.hermeint(c, m=1)
print("First order (m=1):", first_order)

# Second order integration
second_order = H.hermeint(c, m=2)
print("Second order (m=2):", second_order)

# Third order integration
third_order = H.hermeint(c, m=3)
print("Third order (m=3):", third_order)

First order (m=1): [0. 2. 2. 2.]
Second order (m=2): [0. 0. 1. 0.66666667 0.66666667]
Third order (m=3): [0. 0. 0. 0.16666667 0.16666667 0.22222222]

Integration with Scaling

Use the scl parameter to scale results after each integration ?

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

c = np.array([1, 3, 5])

# Integration with scaling factor
scaled_result = H.hermeint(c, scl=2)
print("With scl=2:", scaled_result)

# Compare with normal integration
normal_result = H.hermeint(c)
print("Normal integration:", normal_result)

With scl=2: [0. 2. 3. 3.33333333]
Normal integration: [0. 1. 1.5 1.66666667]

Conclusion

The hermeint() function provides flexible integration of Hermite_e series with customizable integration constants, order, and scaling. Use the k parameter to set integration constants and m for higher-order integration.