Differentiate a Legendre series and multiply each differentiation by a scalar in Python

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

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. If multidimensional, different axes correspond to different variables
• m ? Number of derivatives taken, must be non-negative (Default: 1)
• scl ? Scalar multiplier. Each differentiation is multiplied by scl, resulting in multiplication by scl**m (Default: 1)
• axis ? Axis over which the derivative is taken (Default: 0)

Basic Example

Let's start with a simple example to understand the basic functionality ?

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

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

# Display the array
print("Original coefficients:", coefficients)
print("Shape:", coefficients.shape)

# Differentiate the Legendre series
result = L.legder(coefficients)
print("After differentiation:", result)

Original coefficients: [1 2 3 4]
Shape: (4,)
After differentiation: [ 6.  9. 20.]

Using Scalar Multiplication

Now let's see how the scalar parameter affects the differentiation ?

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

coefficients = np.array([1, 2, 3, 4])

# Differentiate with scalar = -1
result_negative = L.legder(coefficients, scl=-1)
print("With scl = -1:", result_negative)

# Differentiate with scalar = 2
result_double = L.legder(coefficients, scl=2)
print("With scl = 2:", result_double)

# Multiple differentiations with scalar
result_multiple = L.legder(coefficients, m=2, scl=-1)
print("m=2, scl=-1:", result_multiple)

With scl = -1: [ -6.  -9. -20.]
With scl = 2: [12. 18. 40.]
m=2, scl=-1: [18. 80.]

How It Works

The Legendre differentiation follows these rules ?

• Each coefficient is differentiated according to Legendre polynomial rules
• The scalar scl multiplies each differentiation step
• For m derivatives, the final result is multiplied by scl**m
• This is useful for linear changes of variables

Comparison

Conclusion

The legder() function provides flexible Legendre series differentiation with scalar multiplication. Use the scl parameter for linear transformations and m for multiple derivatives.