Differentiate a Hermite_e series and set the derivatives in Python

The Hermite_e series (probabilist's Hermite polynomials) is a mathematical series used in quantum mechanics and probability theory. The weight function is e^(−x²/2). This guide shows how to differentiate Hermite_e series using NumPy's polynomial module.

Formula

The Hermite_e polynomial formula is:

H_n(x) = (−1)^n e^(x²/2) d^n/dx^n(e^(−x²/2))

Where:

• H_n(x) is the nth Hermite polynomial of degree n
• x is the independent variable
• d^n/dx^n denotes the nth derivative with respect to x

Basic Hermite_e Series Differentiation

To differentiate a Hermite_e series, use hermite_e.hermeder() function with coefficient arrays ?

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

# Create coefficient array
coefficients = np.array([1, 2, 3, 4])
print("Coefficients:", coefficients)

# Differentiate once
first_derivative = H.hermeder(coefficients, m=1)
print("First derivative:", first_derivative)

# Differentiate three times
third_derivative = H.hermeder(coefficients, m=3)
print("Third derivative:", third_derivative)

Coefficients: [1 2 3 4]
First derivative: [ 2.  6. 12.]
Third derivative: [24.]

Evaluating Hermite_e Series

Generate and differentiate a Hermite_e series at specific points using hermeval() ?

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

# Define coefficients and evaluation points
coefficients = [-2, -4, 7, -1, 5]
x = np.linspace(-2, 2, 10)

# Evaluate the series
series_values = H.hermeval(x, coefficients)
print("Series values:")
print(series_values)

# Get derivative coefficients
derivative_coeffs = H.hermeder(coefficients, m=2)
print("\nSecond derivative coefficients:", derivative_coeffs)

# Evaluate the second derivative
derivative_values = H.hermeval(x, derivative_coeffs)
print("Second derivative values:")
print(derivative_values)

Series values:
[  549.   207.    63.     7.   -13.   -13.     7.    63.   207.   549.]
Second derivative coefficients: [ 42.  -6.  60.]
Second derivative values:
[ 2520.   954.   460.   138.   -112.   -112.    138.   460.   954.
  2520.]

Working with Multi-dimensional Arrays

For 2D arrays, specify the axis along which to compute derivatives ?

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

# Create 2D coefficient array
coeffs_2d = np.array([[1, 2, 3], [4, 5, 6]])
print("2D coefficients:")
print(coeffs_2d)

# Differentiate along axis 0
deriv_axis0 = H.hermeder(coeffs_2d, m=1, axis=0)
print("\nDerivative along axis 0:")
print(deriv_axis0)

# Differentiate along axis 1
deriv_axis1 = H.hermeder(coeffs_2d, m=1, axis=1)
print("\nDerivative along axis 1:")
print(deriv_axis1)

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

Derivative along axis 0:
[[4. 5. 6.]]

Derivative along axis 1:
[[ 2.  6.]
 [ 5. 12.]]

Parameters

Conclusion

Use hermite_e.hermeder() to differentiate Hermite_e series by specifying the coefficient array and derivative order. The function returns new coefficients representing the differentiated series, which can be evaluated at specific points using hermeval().