5 Best Ways to Integrate a Laguerre Series and Set the Order of Integration in Python

💡 Problem Formulation: When working with Laguerre polynomials, it’s often necessary to integrate them for various applications such as solving differential equations or mathematical modeling. A user looking for ways to integrate a Laguerre series might have a set of coefficients and a function defined by a Laguerre series. They need to find the integral of this function over a specified range and potentially specify the order of the polynomial used in the integration. The desired output is the numerical value of the integral.

Method 1: Using SciPy’s quad Function

SciPy’s quad function is one of the most common methods for numerical integration in Python. This function can be applied to integrate functions given by Laguerre series. The order of the series is implicitly defined by the coefficients. One needs to create a function representing the Laguerre series and then pass it to quad.

Here’s an example:

from scipy.integrate import quad
from scipy.special import eval_genlaguerre
import numpy as np

# Coefficients of the Laguerre series
coefficients = [1, -0.5, 0.25]

# Define the Laguerre series function
def laguerre_series(x, coeffs):
    return sum(c * eval_genlaguerre(i, 0, x) for i, c in enumerate(coeffs))

# Integrate the Laguerre series function
integral, error = quad(laguerre_series, 0, np.inf, args=(coefficients,))

print(integral)

Output: The numerical value of the integral.

This code snippet defines a Laguerre series function using the coefficients provided and then integrates it using SciPy’s quad function from zero to infinity. The eval_genlaguerre function is used to evaluate the generalised Laguerre polynomial of a given order.

Method 2: Using NumPy’s Polynomial Integration

NumPy’s polynomial classes provide a direct method to integrate polynomials. The Laguerre class has an integ method, which can be used to obtain the integral of the Laguerre polynomial. The order of integration is specified by the number of times the integ method is applied.

Here’s an example:

import numpy as np
from numpy.polynomial.laguerre import Laguerre

# Coefficients of the Laguerre polynomial
coefficients = [1, -0.5, 0.25]

# Create a Laguerre object
lag_poly = Laguerre(coefficients)

# Integrate the polynomial
integral_poly = lag_poly.integ()

# Evaluate the integral at the upper limit (infinity)
integral = integral_poly(0)  # Laguerre polynomials are defined for [0, ∞)

print(integral)

Output: The integral of the Laguerre polynomial.

This code example creates a Laguerre object with the given coefficients. The integ method is then called to integrate the polynomial, and the resulting integral is evaluated at zero, which effectively calculates the integral over the whole range.

Method 3: Symbolic Integration with SymPy

SymPy, a Python library for symbolic mathematics, allows symbolic integration of Laguerre polynomials. This can be extremely useful for analytical calculations since the order of integration can be freely adjusted, and the result is an exact symbolic expression.

Here’s an example:

from sympy import symbols, integrate
from sympy.functions.special.polynomials import laguerre

x = symbols('x')
order = 2

# Define the Laguerre polynomial
L = laguerre(order, x)

# Perform symbolic integration
integral = integrate(L, x)

print(integral)

Output: The symbolic expression for the integral of the Laguerre polynomial.

This snippet demonstrates the creation of a Laguerre polynomial of a specific order using SymPy. The function integrate is then used to perform the integration, and the result is a symbolic expression for the integral of the polynomial.

Method 4: Monte Carlo Integration

Monte Carlo integration is a stochastic technique to estimate integrals. This is particularly useful when dealing with high-dimensional integrals or complex domains. While not specific to Laguerre polynomials, this method can be applied to any function, including Laguerre series.

Here’s an example:

import numpy as np
from scipy.special import genlaguerre

# Coefficients of the Laguerre polynomial
coefficients = [1, -0.5, 0.25]

# Define the Laguerre polynomial function
def laguerre_poly(x):
    return np.polynomial.laguerre.lagval(x, coefficients)

# Number of random points for the Monte Carlo simulation
n_points = 10000

# Generate random points
random_points = np.random.gamma(1, 1, n_points)

# Evaluate the Laguerre polynomial at these points
values = laguerre_poly(random_points)

# Estimate the integral
integral = np.mean(values) * np.exp(1)  # Include the weight function e^-x

print(integral)

Output: An estimate of the integral using Monte Carlo simulation.

This code snippet uses a Monte Carlo method to estimate the integral of a Laguerre polynomial. Random points are generated following the exponential distribution, which corresponds to the weight function of the Laguerre polynomials. The function values are then averaged, and the mean is adjusted with the weighting factor to estimate the integral.

Bonus One-Liner Method 5: Using the Quad Function with Lambda

For straightforward integrations where creating a separate function may seem overkill, a lambda function can be used in combination with SciPy’s quad function for a concise one-liner integration of Laguerre series.

Here’s an example:

from scipy.integrate import quad
from scipy.special import eval_genlaguerre
import numpy as np

# Coefficients of the Laguerre series
coefficients = [1, -0.5, 0.25]

# One-liner integration using a lambda function
integral, error = quad(lambda x: sum(c * eval_genlaguerre(i, 0, x) for i, c in enumerate(coefficients)), 0, np.inf)

print(integral)

Output: The numerical value of the integral.

This one-liner uses a lambda function to define the Laguerre series within the call to quad, making it compact and efficient for simple cases.

Summary/Discussion

Method 1:

quad

Method 2:

Method 3:

Method 4:

Bonus One-Liner Method 5: