Differentiate a Legendre series, set the derivatives and multiply each differentiation by a scalar in Python

To differentiate a Legendre series in Python, use the polynomial.legendre.legder() method. This function returns the Legendre series coefficients differentiated m times along the specified axis, with each differentiation multiplied by a scalar value.

Syntax

numpy.polynomial.legendre.legder(c, m=1, scl=1, axis=0)

Parameters

The function accepts the following parameters:

• c − Array of Legendre series coefficients. For multidimensional arrays, different axes correspond to different variables
• m − Number of derivatives to take (must be non-negative, default: 1)
• scl − Scalar multiplier applied to each differentiation (default: 1). Final result is multiplied by scl**m
• axis − Axis along which the derivative is computed (default: 0)

Example

Let's differentiate a Legendre series with custom derivative count and scalar multiplier:

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

# Create an array of coefficients
c = np.array([1, 2, 3, 4])

# 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)

# Differentiate the Legendre series twice with scalar multiplier -1
print("\nResult...")
print(L.legder(c, m=2, scl=-1))

Our Array...
[1 2 3 4]

Dimensions of our Array...
1

Datatype of our Array object...
int64

Shape of our Array object...
(4,)

Result...
[ 9. 60.]

How It Works

In the example above:

• We start with coefficients [1, 2, 3, 4] representing a Legendre series
• The function differentiates twice (m=2)
• Each differentiation is multiplied by scl=-1
• The final result is multiplied by (-1)²=1, but the differentiation process transforms the coefficients

Conclusion

The legder() function provides a convenient way to differentiate Legendre series with custom derivative orders and scalar multipliers. Use the scl parameter for linear variable transformations and m to control the differentiation order.