Question

The bell-shaped Gaussian function, $$ f(x)=\frac{1}{\sqrt{2 \pi} s} \exp \left[-\frac{1}{2}\left(\frac{x-m}{s}\right)^{2}\right] $$ is one of the most widely used functions in science and technology \(^{32}\). The parameters \(m\) and \(s\) are real numbers, where \(s\) must be greater than zero. Make a program for evaluating this function when \(m=0, s=2\), and \(x=1\). Verify the program's result by comparing with hand calculations on a calculator. Name of program file: Gaussian_function1.py.

Short answer

The calculated value of the Gaussian function for \(x=1\), \(m=0\), and \(s=2\) is approximately 0.1760.

Step-by-step solution

Understand the Gaussian Function

The Gaussian function is defined as \( f(x) = \frac{1}{\sqrt{2 \pi} s} \exp\left[-\frac{1}{2}\left(\frac{x-m}{s}\right)^{2}\right] \). In this exercise, we need to evaluate this function for \( m=0 \), \( s=2 \), and \( x=1 \).

Substitute Values into the Function

Plug in the given values into the Gaussian function formula: \( m = 0 \), \( s = 2 \), and \( x = 1 \). The equation becomes \( f(1) = \frac{1}{\sqrt{2 \pi} \cdot 2} \exp\left[-\frac{1}{2}\left(\frac{1-0}{2}\right)^{2}\right] \).

Simplify the Exponent

Calculate the exponent part of the expression: \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \). Therefore the exponent becomes \(-\frac{1}{2} \times \frac{1}{4} = -\frac{1}{8} \).

Evaluate the Exponential Function

Calculate the exponential part: \( \exp(-\frac{1}{8}) \approx 0.8825 \). Use a calculator for this evaluation.

Simplify the Prefactor

Calculate the prefactor: \( \frac{1}{\sqrt{2 \pi} \cdot 2} = \frac{1}{2\sqrt{2\pi}} \approx 0.1995 \). Use a calculator to compute this as well.

Compute the Final Result

Multiply the prefactor and the exponential result: \( f(1) = 0.1995 \times 0.8825 \approx 0.1760 \). This is the value of the Gaussian function for \( x = 1 \), \( m = 0 \), and \( s = 2 \).

Write the Program

Create a Python program in a file named Gaussian_function1.py. Use the 'math' module to compute the Gaussian function for the specified values.

Verify with Hand Calculations

Run the program and verify that the output matches the hand calculation result, which is approximately 0.1760.

Key concepts

Python programming

Creating a program to evaluate a mathematical function like the Gaussian function is a great way to apply Python programming skills. In this exercise, we used Python—a powerful, high-level programming language that's particularly popular for scientific computation.

To write our program, we utilized Python's 'math' module. This module provides mathematical functions that are much more efficient than homemade alternatives.

For example, by using `math.sqrt()` and `math.exp()`, we can easily compute the square root and exponential components of the Gaussian function without needing to write these functions ourselves. Coding our solution involves simple steps:

• Import the 'math' library using `import math`.
• Set variables for given values: `m=0`, `s=2`, and `x=1`.
• Use the formula to compute the result using Python's functions.
• Print the result to verify its accuracy.

This demonstrates not only how to evaluate the Gaussian function programmatically but also showcases the efficiency and simplicity of using Python for numerical calculations.

exponential function

Understanding the exponential function is key in evaluating the Gaussian function. The exponential function, denoted as \( e^x \) or \( \exp(x) \), is fundamental in mathematics because it describes growth or decay processes.

In the Gaussian function, the exponential component is \( \exp\left[-\frac{1}{2}\left(\frac{x-m}{s}\right)^{2}\right] \). This function determines how the value of \( x \) deviates from the mean \( m \) is scaled by \( s \), contributing to the function's characteristic bell shape. The exponential function in this context adjusts the height of the curve by applying a decay factor.

Using Python's `math.exp()` function, we can efficiently calculate exponential values. This allows us to implement exponential decay in the Gaussian function where small deviations close to the mean result in values close to 1, and values far from the mean are closer to 0.

numerical calculation

Numerical calculations are critical for evaluating mathematical functions like the Gaussian function. Hand calculations can be error-prone, especially when dealing with precision and irrational numbers. To avoid this, we use numerical methods to carry out calculations.

When calculating the Gaussian function, both the prefactor (\( \frac{1}{\sqrt{2 \pi} s} \)) and the exponent's value require accurate computation.

Python's `math.sqrt()` and `math.exp()` functionalities help ensure we achieve precise results. Additionally, performing these calculations programmatically allows for consistent and quick evaluations.

To verify the accuracy of our results, hand calculations can also be performed step-by-step to cross-check program outputs, ensuring that our program is working correctly and our computations are accurate.

parameter substitution

Parameter substitution is an effective technique for evaluating functions. It involves replacing variables in an equation with specific values to simplify the problem.

This exercise required substituting the parameters \( m = 0 \), \( s = 2 \), and \( x = 1 \) into the Gaussian function. By doing this:

• The continuous variable \( x \) is evaluated at a specific point.
• Parameters like \( m \) (mean) and \( s \) (standard deviation) define the curve's position and width.

The substitution simplifies the function, allowing us to focus on computation rather than deriving the function shape.

In coding, parameter substitution is implemented through assigning values to variables, making it easy to modify or navigate through different scenarios of a function. Whether in calculations or programming, parameter substitution is a fundamental concept that underpins evaluating function values effectively.