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

To differentiate a Laguerre series, use the laguerre.lagder() method in Python. The method returns the Laguerre series coefficients differentiated m times along axis. At each iteration, the result is multiplied by a scalar value scl.

The coefficient array c represents coefficients from low to high degree. For example, [1,2,3] represents the series 1*L_0 + 2*L_1 + 3*L_2, where L_n are Laguerre polynomials.

Syntax

numpy.polynomial.laguerre.lagder(c, m=1, scl=1, axis=0)

Parameters

• c ? Array of Laguerre series coefficients
• m ? Number of derivatives taken (default: 1)
• scl ? Scalar multiplier applied to each differentiation (default: 1)
• axis ? Axis over which derivative is taken (default: 0)

Example

Let's differentiate a Laguerre series and multiply by a scalar ?

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

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

# Display the original array
print("Original coefficients:")
print(c)

# Differentiate once with scalar -1
result1 = L.lagder(c, scl=-1)
print("\nFirst derivative (scl=-1):")
print(result1)

# Differentiate twice with scalar 2
result2 = L.lagder(c, m=2, scl=2)
print("\nSecond derivative (scl=2):")
print(result2)

Original coefficients:
[1 2 3 4]

First derivative (scl=-1):
[-2. -4. -4.]

Second derivative (scl=2):
[-8. -8.]

How It Works

The differentiation process applies the derivative operation to each Laguerre polynomial term and multiplies by the scalar. For multiple derivatives (m > 1), the final result is multiplied by scl^m.

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

# Example with different scalar values
coeffs = np.array([1, 3, 5])

print("Original coefficients:", coeffs)
print("Derivative (scl=1):", L.lagder(coeffs, scl=1))
print("Derivative (scl=3):", L.lagder(coeffs, scl=3))
print("Second derivative (scl=2):", L.lagder(coeffs, m=2, scl=2))

Original coefficients: [1 3 5]
Derivative (scl=1): [-2.  2.]
Derivative (scl=3): [-6.  6.]
Second derivative (scl=2): [16.]

Conclusion

The laguerre.lagder() method efficiently differentiates Laguerre series and applies scalar multiplication. Use the scl parameter to scale derivatives and m parameter for higher-order derivatives.