Integrate a Legendre series and set the lower bound of the integral in Python

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

Syntax

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

Parameters

Example

Let's integrate a Legendre series with coefficients [1, 2, 3] and set the lower bound to -2 ?

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

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

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

# Check the dimensions
print("\nDimensions of our Array...")
print(c.ndim)

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

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

# Integrate the Legendre series with lower bound = -2
result = L.legint(c, lbnd=-2)
print("\nResult...")
print(result)

Our Array...
[1 2 3]

Dimensions of our Array...
1

Datatype of our Array object...
int64

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

Result...
[7.33333333 0.4        0.66666667 0.6       ]

How It Works

When integrating a Legendre series with lbnd=-2:

• The original series has coefficients [1, 2, 3] representing: 1 + 2P?(x) + 3P?(x)
• Integration increases the degree by 1, resulting in 4 coefficients
• The lower bound lbnd=-2 determines the constant of integration
• Each coefficient is adjusted based on the integration rules for Legendre polynomials

Different Lower Bounds

Let's see how different lower bounds affect the result ?

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

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

# Different lower bounds
bounds = [0, -1, -2, 1]

for bound in bounds:
    result = L.legint(c, lbnd=bound)
    print(f"lbnd = {bound}: {result}")

lbnd = 0: [-1.33333333  0.4         0.66666667  0.6       ]
lbnd = -1: [ 3.          0.4         0.66666667  0.6       ]
lbnd = -2: [7.33333333 0.4        0.66666667 0.6       ]
lbnd = 1: [-5.66666667  0.4         0.66666667  0.6       ]

Conclusion

The legint() method integrates Legendre series coefficients, with the lbnd parameter controlling the lower bound of integration. Different lower bounds produce different constant terms while preserving the higher-order coefficients.