Integrate a Laguerre series and multiply the result by a scalar before the integration constant is added in Python

To integrate a Laguerre series and multiply by a scalar, use the numpy.polynomial.laguerre.lagint() method in Python. This method integrates the Laguerre series coefficients and applies a scaling factor before adding the integration constant.

Syntax

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

Parameters

The key parameters are ?

• c ? Array of Laguerre series coefficients
• m ? Order of integration (default: 1)
• k ? Integration constant(s) (default: [])
• lbnd ? Lower bound of integration (default: 0)
• scl ? Scaling factor applied after each integration (default: 1)
• axis ? Axis over which to integrate (default: 0)

Basic Integration Example

Let's start with basic Laguerre series integration ?

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

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

# Basic integration
result = L.lagint(c)
print("Integrated coefficients:", result)

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

Integration with Scaling Factor

Apply a scaling factor during integration ?

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

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

# Integration with scaling factor -2
result = L.lagint(c, scl=-2)
print("Result with scl=-2:", result)

# Integration with scaling factor 0.5
result2 = L.lagint(c, scl=0.5)
print("Result with scl=0.5:", result2)

Original coefficients: [1 2 3]
Result with scl=-2: [-2. -2. -2.  6.]
Result with scl=0.5: [ 0.5  0.5  0.5 -0.5]

Multiple Integration Orders

Integrate multiple times with different parameters ?

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

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

# Integration with order m=2 and scaling
result_m2 = L.lagint(c, m=2, scl=2)
print("Double integration (m=2, scl=2):", result_m2)

# Integration with custom integration constant
result_k = L.lagint(c, k=[5], scl=-1)
print("With integration constant k=5:", result_k)

Original coefficients: [1 2 3]
Double integration (m=2, scl=2): [ 4.  4.  4. -4. -2.]
With integration constant k=5: [ 4.  1.  1. -1.]

How It Works

The integration process follows these steps ?

1. Integrate the Laguerre series coefficients
2. Multiply the result by the scaling factor scl
3. Add the integration constant k
4. Repeat for m iterations if multiple integration is specified

Comparison

Conclusion

Use lagint() with the scl parameter to integrate Laguerre series and apply scaling. The scaling factor multiplies the result after each integration step, allowing for linear transformations during the integration process.