Integrate a Laguerre series and set the Integration constant in Python

To integrate a Laguerre series, use the laguerre.lagint() method in Python. The method returns the Laguerre series coefficients c integrated m times from lbnd along axis. At each iteration the resulting series is multiplied by scl and an integration constant, k, is added.

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. If multidimensional, different axes correspond to different variables
• m ? Order of integration, must be positive (Default: 1)
• k ? Integration constant(s). If k == [] (default), all constants are set to zero. For m == 1, a single scalar can be given
• 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)

Example

Let's create a Laguerre series and integrate it with different integration constants ?

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

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

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

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

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

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

# Integrate with integration constant k = 3
print("\nResult with k=3...\n", L.lagint(c, k=3))

Our Array...
 [1 2 3]

Dimensions of our Array...
1

Datatype of our Array object...
int64

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

Result with k=3...
 [ 4.  1.  1. -3.]

Integration with Different Constants

Let's see how different integration constants affect the result ?

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

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

# Default integration (k=0)
print("Default integration (k=0):", L.lagint(c))

# Integration with k=5
print("Integration with k=5:", L.lagint(c, k=5))

# Integration with k=-2
print("Integration with k=-2:", L.lagint(c, k=-2))

Default integration (k=0): [ 1.  1.  1. -3.]
Integration with k=5: [ 6.  1.  1. -3.]
Integration with k=-2: [-1.  1.  1. -3.]

Multiple Integration

You can also perform multiple integrations by setting the m parameter ?

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

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

# Single integration
print("Single integration:", L.lagint(c, m=1, k=1))

# Double integration with multiple constants
print("Double integration:", L.lagint(c, m=2, k=[1, 2]))

Single integration: [ 2.  1.  1. -3.]
Double integration: [ 3.  2.  0.5  0.5 -0.75]

Conclusion

The laguerre.lagint() method integrates Laguerre series with customizable integration constants. The integration constant k directly affects the first coefficient of the result, while higher-order coefficients depend on the original series.