Differentiate a Hermite_e series, set the derivatives and multiply each differentiation by a scalar in Python

To differentiate a Hermite_e series, use the hermite_e.hermeder() method in Python. This function allows you to compute derivatives and apply scalar multiplication to each differentiation step.

Syntax

numpy.polynomial.hermite_e.hermeder(c, m=1, scl=1, axis=0)

Parameters

The function accepts the following parameters ?

• c ? Array of Hermite_e series coefficients. For multidimensional arrays, different axes correspond to different variables
• m ? Number of derivatives to take (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

Let's create a Hermite_e series and compute its second derivative with scalar multiplication ?

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

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

# Display the array
print("Our Array...")
print(c)

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

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

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

# To differentiate a Hermite_e series, use the hermite_e.hermeder() method
# Taking 2nd derivative with scalar multiplier -1
print("\nResult...")
print(H.hermeder(c, 2, 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...
[ 6. 24.]

How It Works

The original polynomial represented by coefficients [1, 2, 3, 4] corresponds to:

P(x) = 1 + 2x + 3x² + 4x³

Taking the second derivative and multiplying by scl = -1 twice (giving (-1)² = 1), we get coefficients [6, 24], representing 6 + 24x.

Multiple Derivatives with Different Scalars

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

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

# First derivative
first_deriv = H.hermeder(c, m=1)
print("First derivative:", first_deriv)

# Second derivative with scalar 2
second_deriv = H.hermeder(c, m=2, scl=2)
print("Second derivative (scl=2):", second_deriv)

# Third derivative with scalar -1
third_deriv = H.hermeder(c, m=3, scl=-1)
print("Third derivative (scl=-1):", third_deriv)

First derivative: [ 2.  6. 12. 20.]
Second derivative (scl=2): [24. 96. 320.]
Third derivative (scl=-1): [-144. -1280.]

Conclusion

The hermite_e.hermeder() method efficiently computes derivatives of Hermite_e series with optional scalar multiplication. Use the scl parameter for linear variable transformations and m to specify the derivative order.