Integrate a Hermite series and set the Integration constant in Python

To integrate a Hermite series, use the hermite.hermint() method in Python. This method integrates a Hermite series and allows you to set integration constants.

Syntax

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

Parameters

The method accepts the following 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. For m=1, can be a scalar
• 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)

Basic Integration Example

Let's integrate a simple Hermite series without setting integration constants ?

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

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

# Integrate without setting integration constant
result = H.hermint(c)
print("Integrated series:", result)

Original coefficients: [1 2 3]
Integrated series: [0.  0.5 0.5 0.5]

Setting Integration Constants

Now let's integrate the same series but set the integration constant to 3 ?

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

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

# Integrate with integration constant k=3
result = H.hermint(c, k=3)
print("With integration constant k=3:", result)

# Multiple integration with different constants
result_multi = H.hermint(c, m=2, k=[1, 2])
print("Double integration with k=[1, 2]:", result_multi)

Original coefficients: [1 2 3]
With integration constant k=3: [3.  0.5 0.5 0.5]
Double integration with k=[1, 2]: [1.    2.    0.125 0.125 0.125]

Integration Order and Bounds

You can control the order of integration and set different bounds ?

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

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

# Second order integration
result_order2 = H.hermint(c, m=2)
print("Second order integration:", result_order2)

# Integration with different lower bound
result_bound = H.hermint(c, lbnd=1)
print("With lbnd=1:", result_bound)

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

Second order integration: [0.    0.    0.125 0.125 0.125]
With lbnd=1: [0.  0.5 0.5 0.5]
With scaling factor scl=2: [0. 1. 1. 1.]

Conclusion

The hermite.hermint() method provides flexible integration of Hermite series with customizable integration constants. Use the k parameter to set constants, m for integration order, and scl for scaling factors.