Integrate a Hermite series in Python

To integrate a Hermite series in Python, use the hermite.hermint() method from NumPy. This function performs polynomial integration on Hermite series coefficients and returns the integrated coefficients.

Syntax

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

Parameters

• c ? Array of Hermite series coefficients. For multidimensional arrays, different axes correspond to different variables
• m ? Order of integration (must be positive, default: 1)
• k ? Integration constant(s). If empty list (default), all constants are zero
• lbnd ? Lower bound of the integral (default: 0)
• scl ? Scalar multiplier applied after each integration (default: 1)
• axis ? Axis over which integration is performed (default: 0)

Example

Let's create a simple Hermite series and integrate it ?

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

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

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

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

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

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

# To integrate a Hermite series, use the hermite.hermint() method
print("\nResult...")
print(H.hermint(c))

Our Array...
[1 2 3]

Dimensions of our Array...
1

Datatype of our Array object...
int64

Shape of our Array object...
(3,)

Result...
[1.  0.5 0.5 0.5]

Integration with Different Parameters

You can also specify integration order and constants ?

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

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

# Integrate twice (m=2) with integration constant k=5
result = H.hermint(c, m=2, k=5)
print("Integration with m=2, k=5:")
print(result)

# Integration with scaling factor
result_scaled = H.hermint(c, scl=2)
print("\nIntegration with scaling factor scl=2:")
print(result_scaled)

Integration with m=2, k=5:
[5.    1.    0.25  0.125 0.125]

Integration with scaling factor scl=2:
[2. 1. 1. 1.]

How It Works

The hermint() function integrates each term of the Hermite series. For a coefficient array [1, 2, 3] representing the series 1?H?(x) + 2?H?(x) + 3?H?(x), integration produces a new series with an additional term and modified coefficients.

Conclusion

The hermite.hermint() method provides a convenient way to integrate Hermite series in Python. Use the m parameter for multiple integrations and k for integration constants.