Integrate a Laguerre series over specific axis in Python

The Laguerre polynomial integration in Python can be performed using NumPy's laguerre.lagint() method. This function integrates Laguerre series coefficients along a specified axis, allowing for flexible multi-dimensional operations.

Syntax

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

Parameters

The function accepts the following parameters:

• c − Array of Laguerre series coefficients
• m − Order of integration (default: 1)
• k − Integration constant(s) (default: [])
• lbnd − Lower bound of integral (default: 0)
• scl − Scaling factor (default: 1)
• axis − Axis over which integration is performed (default: 0)

Basic Integration Example

Let's start with a simple example of integrating a Laguerre series ?

import numpy as np
from numpy.polynomial import laguerre as L

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

# Integrate the Laguerre series
result = L.lagint(coefficients)
print("Integrated coefficients:", result)

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

Integration Along Specific Axis

For multidimensional arrays, you can specify which axis to integrate along ?

import numpy as np
from numpy.polynomial import laguerre as L

# Create a multidimensional array of coefficients
c = np.arange(4).reshape(2, 2)
print("Our Array...")
print(c)

print("\nDimensions:", c.ndim)
print("Shape:", c.shape)
print("Datatype:", c.dtype)

# Integrate along axis 1
result = L.lagint(c, axis=1)
print("\nResult (axis=1):")
print(result)

Our Array...
[[0 1]
 [2 3]]

Dimensions: 2
Shape: (2, 2)
Datatype: int64

Result (axis=1):
[[ 0.  1. -1.]
 [ 2.  1. -3.]]

Integration with Custom Parameters

You can control the integration with additional parameters ?

import numpy as np
from numpy.polynomial import laguerre as L

# Create coefficient array
coefficients = np.array([1, 2, 1])

# Integrate with custom parameters
result1 = L.lagint(coefficients, m=1, k=[5])  # Integration constant = 5
result2 = L.lagint(coefficients, m=2)         # Second order integration

print("Original coefficients:", coefficients)
print("Integration with k=[5]:", result1)
print("Second order integration:", result2)

Original coefficients: [1 2 1]
Integration with k=[5]: [ 6.  1.  0.  1.]
Second order integration: [ 1.   1.  -1.   0.   1.]

Comparison

Conclusion

The laguerre.lagint() function provides flexible integration of Laguerre series with control over integration order, constants, and axis selection. This is particularly useful for solving differential equations and mathematical modeling involving Laguerre polynomials.