Multiply a Legendre series by an independent variable in Python

To multiply a Legendre series by an independent variable x, use the polynomial.legendre.legmulx() method in NumPy. This method takes a 1-D array of Legendre series coefficients and returns the result of multiplying the series by x.

Syntax

numpy.polynomial.legendre.legmulx(c)

Parameters

The parameter c is a 1-D array of Legendre series coefficients ordered from low to high degree. For example, [1, 2, 3] represents the series P? + 2×P? + 3×P?, where P? are Legendre polynomials.

Example

Let's create a Legendre series and multiply it by x ?

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

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

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

# Check the dimensions
print("\nDimensions:", c.ndim)

# Get the datatype
print("Datatype:", c.dtype)

# Get the shape
print("Shape:", c.shape)

# Multiply the Legendre series by x
result = L.legmulx(c)
print("\nResult after multiplying by x:")
print(result)

Our Array:
[1 2 3]

Dimensions: 1
Datatype: int64
Shape: (3,)

Result after multiplying by x:
[0.66666667 2.2        1.33333333 1.8       ]

How It Works

The legmulx() method multiplies each Legendre polynomial in the series by the independent variable x. This operation increases the degree of each term by 1, resulting in a series with one additional coefficient.

The original series [1, 2, 3] represents:

• 1×P? + 2×P? + 3×P?

After multiplication by x, we get a series of degree 3 with 4 coefficients representing the new polynomial.

Conclusion

Use numpy.polynomial.legendre.legmulx() to multiply a Legendre series by the independent variable x. The method returns a new array with coefficients representing the multiplied series, increasing the polynomial degree by one.