Integrate a Legendre series over specific axis in Python

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

Syntax

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

Parameters

The method accepts the following parameters ?

• c − Array of Legendre series coefficients. If multidimensional, 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 the integral is taken (Default: 0)

Example

Let's create a multidimensional array and integrate along different axes ?

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

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

# 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 along axis 1
print("\nIntegration along axis=1...\n", L.legint(c, axis=1))

# Integrate along axis 0
print("\nIntegration along axis=0...\n", L.legint(c, axis=0))

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

Dimensions of our Array...
 2

Datatype of our Array object...
 int64

Shape of our Array object...
 (2, 2)

Integration along axis=1...
 [[0.16666667 0.         0.33333333]
 [0.5        2.         1.        ]]

Integration along axis=0...
 [[0.         0.5       ]
 [0.66666667 0.        ]
 [0.66666667 1.        ]]

Integration with Custom Parameters

You can also specify integration order, constants, and bounds ?

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

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

# Single integration (default)
result1 = L.legint(c)
print("Single integration:\n", result1)

# Double integration with integration constant
result2 = L.legint(c, m=2, k=[1, 2])
print("\nDouble integration with constants:\n", result2)

# Integration with scaling factor
result3 = L.legint(c, scl=2, lbnd=1)
print("\nIntegration with scaling and bound:\n", result3)

Single integration:
 [0.33333333 0.         0.66666667 0.         1.        ]

Double integration with constants:
 [ 1.06666667  2.          0.11111111  0.          0.22222222  0.
   0.33333333]

Integration with scaling and bound:
 [-0.33333333  0.          1.33333333  0.          2.        ]

Conclusion

The legint() method provides flexible Legendre series integration with control over axis, order, constants, and scaling. Use the axis parameter to integrate along specific dimensions of multidimensional coefficient arrays.