Differentiate a Laguerre series and set the derivatives 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 the specified axis. At each iteration, the result is multiplied by a scaling factor.

The argument c is an array of coefficients from low to high degree along each axis. For example, [1,2,3] represents the series 1*L? + 2*L? + 3*L?, while [[1,2],[1,2]] represents a two-dimensional series if axis=0 is x and axis=1 is y.

Syntax

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

Parameters

Basic Differentiation Example

Let's start with a simple example of differentiating a Laguerre series ?

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

# Create coefficients for series: 1*L_0 + 2*L_1 + 3*L_2 + 4*L_3
c = np.array([1, 2, 3, 4])
print("Original coefficients:", c)

# First derivative
result1 = L.lagder(c, m=1)
print("First derivative:", result1)

# Second derivative
result2 = L.lagder(c, m=2)
print("Second derivative:", result2)

# Third derivative
result3 = L.lagder(c, m=3)
print("Third derivative:", result3)

Original coefficients: [1 2 3 4]
First derivative: [ 1.  1.  1.]
Second derivative: [ 0. -2.]
Third derivative: [-4.]

Using Scaling Factor

The scl parameter multiplies each differentiation by a scalar value ?

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

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

# Differentiate with scaling factor of 2
result_scaled = L.lagder(c, m=1, scl=2)
print("With scl=2:", result_scaled)

# Compare with normal differentiation
result_normal = L.lagder(c, m=1)
print("Normal:", result_normal)
print("Manual scaling:", result_normal * 2)

With scl=2: [2. 2. 2.]
Normal: [1. 1. 1.]
Manual scaling: [2. 2. 2.]

Multidimensional Arrays

For multidimensional coefficient arrays, specify the axis for differentiation ?

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

# Create a 2D array of coefficients
c_2d = np.array([[1, 2], [3, 4], [5, 6]])
print("2D coefficients:")
print(c_2d)

# Differentiate along axis 0 (rows)
result_axis0 = L.lagder(c_2d, axis=0)
print("\nDifferentiate along axis 0:")
print(result_axis0)

# Differentiate along axis 1 (columns)  
result_axis1 = L.lagder(c_2d, axis=1)
print("\nDifferentiate along axis 1:")
print(result_axis1)

2D coefficients:
[[1 2]
 [3 4]
 [5 6]]

Differentiate along axis 0:
[[ 2.  2.]
 [ 2.  2.]]

Differentiate along axis 1:
[[1.]
 [1.]
 [1.]]

Complete Example

Here's a comprehensive example showing array properties and differentiation ?

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

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

print("Our Array...\n", c)
print("\nDimensions of our Array...\n", c.ndim)
print("\nDatatype of our Array object...\n", c.dtype)  
print("\nShape of our Array object...\n", c.shape)

# Differentiate the Laguerre series three times
print("\nThird derivative result...\n", L.lagder(c, 3))

Our Array...
 [1 2 3 4]

Dimensions of our Array...
 1

Datatype of our Array object...
 int64

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

Third derivative result...
 [-4.]

Conclusion

The laguerre.lagder() method efficiently differentiates Laguerre series with customizable order, scaling, and axis selection. Use the m parameter to control derivative order and scl for linear transformations.