Differentiate a Hermite_e series in Python

To differentiate a Hermite_e series, use the hermite_e.hermeder() method in Python. This function computes the derivative of a Hermite_e polynomial series represented by its coefficients.

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. If multidimensional, different axes correspond to different variables
• m − Number of derivatives to take (default: 1). Must be non-negative
• scl − Scalar multiplier for each differentiation (default: 1)
• axis − Axis over which the derivative is taken (default: 0)

Example

Let's differentiate a Hermite_e series with coefficients [1, 2, 3, 4] ?

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

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

# Differentiate the Hermite_e series
derivative = H.hermeder(c)
print("First derivative:", derivative)

# Second derivative
second_derivative = H.hermeder(c, m=2)
print("Second derivative:", second_derivative)

Original coefficients: [1 2 3 4]
First derivative: [ 2.  6. 12.]
Second derivative: [ 6. 24.]

Multiple Derivatives

You can compute higher-order derivatives by specifying the m parameter ?

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

c = np.array([1, 2, 3, 4, 5])
print("Original coefficients:", c)

# First, second, and third derivatives
for order in range(1, 4):
    derivative = H.hermeder(c, m=order)
    print(f"Derivative order {order}:", derivative)

Original coefficients: [1 2 3 4 5]
Derivative order 1: [ 2.  6. 12. 20.]
Derivative order 2: [ 6. 24. 60.]
Derivative order 3: [24. 120.]

Using Scale Parameter

The scl parameter multiplies each differentiation step ?

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

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

# Without scaling
normal = H.hermeder(c)
print("Normal derivative:", normal)

# With scaling factor of 2
scaled = H.hermeder(c, scl=2)
print("Scaled derivative (scl=2):", scaled)

Normal derivative: [ 2.  6. 12.]
Scaled derivative (scl=2): [ 4. 12. 24.]

Conclusion

The hermite_e.hermeder() function provides efficient differentiation of Hermite_e polynomial series. Use the m parameter for higher-order derivatives and scl for linear transformations during differentiation.