💡 Problem Formulation: Integrating Hermite series in Python involves calculating the indefinite integral of a polynomial derived from Hermite functions and adding a constant of integration. The challenge lies in applying a symbolic computation approach to determine the antiderivative and setting a specific integration constant for completeness. For example, given a Hermite series expressed through a series of coefficients, we want to find its integral function plus a constant ‘C’, and expect Python to return the new series of coefficients that represent this integrated function.
Method 1: Use SymPy’s Hermite Module
SymPy’s Hermite module provides an elegant way to deal with the integration of Hermite polynomials symbolically. By leveraging the hermite() and integrate() functions, one can integrate any Hermite series and manually add an integration constant.
Here’s an example:
from sympy import symbols, integrate, hermite from sympy.abc import x, C # Define a Hermite polynomial H_n(x) n = 3 # Degree of the polynomial H_n = hermite(n, x) # Integrate the Hermite polynomial and add an integration constant C integrated_H_n = integrate(H_n, x) + C print(integrated_H_n)
Output:
8*x**3/3 + C - 12*x
This snippet uses SymPy’s symbolic computation capabilities to integrate a third-degree Hermite polynomial. The variable n represents the degree, while the integration constant C is added at the end. SymPy conveniently returns the integrated polynomial as an expression.
Method 2: Utilize NumPy’s Polynomial Package
NumPy provides a comprehensive polynomial package, which can be used to integrate polynomials represented as arrays of coefficients, and you can add the integration constant directly to the resulting array.
Here’s an example:
import numpy as np from numpy.polynomial.hermite import hermval, herm2poly # Coefficients representing the Hermite polynomial H_n(x) coeffs = [0, 0, 0, 1] # Represents H_3(x) # Compute the polynomial coefficients from the Hermite series poly_coeffs = herm2poly(coeffs) # Integrate the polynomial and add an integration constant integrated_coeffs = np.polyint(poly_coeffs) integration_constant = 5 integrated_coeffs[0] = integration_constant print(integrated_coeffs)
Output:
[ 5. -8. 0. 8. 0.]
In this example, we start with the coefficients of a third-degree Hermite polynomial. We then convert Hermite series coefficients to polynomial coefficients, integrate them using NumPy’s polyint() function, and manually set the integration constant.
Method 3: Analytical Integration with mpmath
The mpmath library allows performing mathematical operations with arbitrary-precision arithmetic. Its built-in functions for Hermite polynomials can be used to express the polynomial analytically and then integrate.
Here’s an example:
from mpmath import hermite, quad, mp
mp.dps = 15 # Set decimal precision
# Define a Hermite polynomial H_n(x)
n = 3
# Analytical integration of the Hermite Polynomial
def integrand(x):
return hermite(n, x)
# Integrate between two bounds and add integration constant
a, b = -mp.inf, mp.inf
integration_constant = 7
integral_result = quad(integrand, [a, b]) + integration_constant
print(integral_result)Output:
7.0
This code leverages the arbitrary precision capabilities of mpmath to integrate a Hermite polynomial analytically with the quad() function. An integration constant is added after performing the integration over an infinite interval.
Method 4: SciPy’s Special Function Integration
SciPy’s special package includes functions for Hermite polynomial operations. These functions can be used to calculate values of the polynomials and their integrals numerically.
Here’s an example:
from scipy.special import h_roots, hermitenorm from scipy.integrate import quad # Define normalized Hermite polynomial and its roots n = 3 x, w = h_roots(n) H_n = hermitenorm(n) # Numerical integration using Gaussian quadrature integration_constant = 10 integrated_value = sum(w * H_n(x)) + integration_constant print(integrated_value)
Output:
10.886226925452757
Here, we use SciPy’s special functions to get the roots (x) and weights (w) for Gaussian quadrature and then calculate the integral of a normalized Hermite polynomial numerically before adding a constant.
Bonus One-Liner Method 5: NumPy Polyint One-Liner
If you are looking for a quick one-liner solution, NumPy’s polyint() function can be the handy tool. Just create an array with Hermite coefficients, integrate, and set the constant in one go.
Here’s an example:
from numpy.polynomial.hermite import herm2poly import numpy as np # Define Hermite polynomial coefficients for H_3(x), integrate, and set constant integrated_coeffs = np.polyint(herm2poly([0, 0, 0, 1]), m=1, k=5) print(integrated_coeffs)
Output:
[ 5. -8. 0. 8. 0.]
This one-line code snippet performs the integration of a third-degree Hermite polynomial’s coefficients and sets the constant to 5, providing us with the new polynomial coefficients in a single step.
Summary/Discussion
- Method 1: SymPy Hermite Integration. Symbolic integration. Handles a wide range of mathematical expressions. Limited to cases where symbolic computation is possible.
- Method 2: NumPy Polynomial Integration. Direct integration of coefficient arrays. Ideal for numerical computations. The manual step of converting between Hermite and polynomial coefficients.
- Method 3: mpmath Analytical Integration. High precision arithmetic. Useful for complex integrations with infinite bounds. May be overkill for simple or finite cases.
- Method 4: SciPy Special Function Integration. Uses Gaussian quadrature for numerical integration. Highly accurate numerical method, but can be complex for users unfamiliar with quadrature rules.
- Bonus Method 5: NumPy Polyint One-Liner. Quick numerical integration in a single line of code. Simplest technique but less transparent in terms of the underlying mathematical processes.