Differentiate a Laguerre series 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 scl.

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_0 + 2*L_1 + 3*L_2, while [[1,2],[1,2]] represents 1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y) if axis=0 is x and axis=1 is y.

Syntax

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

Parameters

• c: Array of Laguerre series coefficients. If multidimensional, different axes correspond to different variables.
• m: Number of derivatives taken, must be non-negative (Default: 1)
• scl: Scalar multiplier for each differentiation. Final result is multiplied by scl**m (Default: 1)
• axis: Axis over which the derivative is taken (Default: 0)

Example

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 array
print("Our Array...\n", c)

# Check the Dimensions
print("\nDimensions of our Array...\n", c.ndim)

# Get the Datatype
print("\nDatatype of our Array object...\n", c.dtype)

# Get the Shape
print("\nShape of our Array object...\n", c.shape)

# To differentiate a Laguerre series, use the laguerre.lagder() method
print("\nResult...\n", L.lagder(c))

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. -7. -4.]

Multiple Derivatives

You can take multiple derivatives by specifying the m parameter:

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

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

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

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

First derivative: [-9. -7. -4.]
Second derivative: [14. 12.]

Conclusion

The laguerre.lagder() method efficiently differentiates Laguerre series by taking derivatives of the coefficient array. Use the m parameter to control the number of derivatives and scl for scaling operations.