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

To differentiate a Hermite_e series and multiply each differentiation by a scalar, use the hermite_e.hermeder() method in Python. This function computes derivatives of Hermite_e series with optional scalar multiplication.

Syntax

numpy.polynomial.hermite_e.hermeder(c, m=1, scl=1, axis=-1)

Parameters

The function takes the following parameters:

• c: Array of Hermite_e series coefficients. For multidimensional arrays, 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 this value (Default: 1)
• axis: Axis over which the derivative is taken (Default: -1)

Example

Let's differentiate a Hermite_e series with scalar multiplication ?

import numpy as np
from numpy.polynomial import hermite_e as H

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

# Display the array
print("Our Array...\n", coefficients)

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

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

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

# Differentiate the Hermite_e series with scalar multiplication
print("\nResult with scl = -1...\n", H.hermeder(coefficients, 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 with scl = -1...
 [ -2.  -6. -12.]

Multiple Derivatives

You can take multiple derivatives and see the effect of scalar multiplication ?

import numpy as np
from numpy.polynomial import hermite_e as H

coefficients = np.array([1, 2, 3, 4, 5])

print("Original coefficients:", coefficients)
print("First derivative (scl=1):", H.hermeder(coefficients, m=1, scl=1))
print("First derivative (scl=2):", H.hermeder(coefficients, m=1, scl=2))
print("Second derivative (scl=2):", H.hermeder(coefficients, m=2, scl=2))

Original coefficients: [1 2 3 4 5]
First derivative (scl=1): [ 2.  6. 12. 20.]
First derivative (scl=2): [ 4. 12. 24. 40.]
Second derivative (scl=2): [12. 48. 80.]

How It Works

For a Hermite_e polynomial represented by coefficients [c?, c?, c?, ..., c?], the derivative removes the first coefficient and multiplies the remaining ones by their respective indices. The scalar multiplication affects each derivative calculation.

Conclusion

The hermite_e.hermeder() method efficiently computes derivatives of Hermite_e series with scalar multiplication. Use the scl parameter to apply linear transformations during differentiation.