The Look

I was quite impressed with the look of the graph I got from using Merlin's text, yesterday. 

Decided to learn from the code...

You

numpy; are single quotes different from double quotes?

Copilot

In Python, including when using libraries like NumPy, single quotes (') and double quotes (") are generally interchangeable for defining string literals. However, there are some nuances to consider:

Consistency: It’s a good practice to be consistent with your choice of quotes within a project. This helps improve readability and maintainability of your code.

Escaping Characters: If your string contains a quote character, using the opposite type of quote can make your code cleaner. For example:

Single quotes inside double quotes: print("It's a beautiful day!")

Double quotes inside single quotes: print('He said, "Hello!"')

Triple Quotes: For multi-line strings or strings that contain both single and double quotes, you can use triple quotes (''' or """):

Python

long_string = """This is a string

that spans multiple lines

and contains both 'single' and "double" quotes."""

In summary, while both single and double quotes can be used interchangeably, choosing one over the other can help avoid the need for escaping characters and improve code readability.

You

What does using the dollat sign before and after a math expression do?

Copilot

Using dollar signs before and after a math expression is a way to format mathematical expressions in LaTeX, a typesetting system commonly used for technical and scientific documentation. Here’s how it works:

Inline Math: When you wrap a math expression in single dollar signs ($...$), it will be formatted inline with the text. For example, $a^2 + b^2 = c^2$ will display as (a^2 + b^2 = c^2).

Display Math: When you wrap a math expression in double dollar signs ($$...$$), it will be formatted as a separate block, centered on its own line. For example, $$a^2 + b^2 = c^2$$ will display as:

a2+b2=c2

This distinction helps in making complex mathematical expressions more readable and properly formatted in documents.

You

What does \ do in a math expression text?

Copilot

In LaTeX, the backslash (\) is used as an escape character to denote commands and special symbols. Here are a few common uses:

Commands: Many LaTeX commands start with a backslash. For example, \frac{a}{b} creates a fraction, and \sqrt{x} produces a square root.

Special Characters: The backslash is used to include special characters that would otherwise be interpreted as commands. For example, \$ displays a dollar sign, \% displays a percent sign, and \_ displays an underscore.

Newlines and Spaces: The backslash can also be used to create newlines (\\) or to add extra space (\quad for a large space, \, for a small space).

Here’s a simple example:

E = mc^2 \quad \text{(Einstein's equation)}

This will display as:

                                                                      E=mc2   (Einstein’s equation)

                                                               *     *     *

Reconstructed my original function from the Transform. The Python code is from Perplexity ai. 

import numpy as np

import matplotlib.pyplot as plt

from scipy.fft import fft, ifft

# Define the original function f(t)

def f(t):

    return np.exp(-t**2)

# Parameters

t_min = -5

t_max = 5

num_points = 1024

# Generate time variable

t = np.linspace(t_min, t_max, num_points)

dt = t[1] - t[0]

# Compute the Fourier Transform of f(t)

F = fft(f(t))

# Frequency variable

frequencies = np.fft.fftfreq(num_points, dt)

# Compute the Inverse Fourier Transform

f_reconstructed = ifft(F)

# Plotting the results

plt.figure(figsize=(12, 6))

# Plot original function

plt.subplot(1, 2, 1)

plt.plot(t, f(t), label='Original f(t) = exp(-t^2)')

plt.title('Original Function')

plt.xlabel('t')

plt.ylabel('f(t)')

plt.grid()

plt.legend()

# Plot reconstructed function

plt.subplot(1, 2, 2)

plt.plot(t, f_reconstructed.real, label='Reconstructed f(t) (Inverse FT)', color='orange')

plt.title('Reconstructed Function from Inverse Fourier Transform')

plt.xlabel('t')

plt.ylabel('f(t)')

plt.grid()

plt.legend()

plt.tight_layout()

plt.show()

                                                                   

                                    

                                                                *     *     *

With Matlab:

                                                                    

% Define parameters N = 1000; % Number of points t_max = 5; % Maximum time value dt = 2*t_max / (N-1); t = linspace(-t_max, t_max, N); % Define the Gaussian function f = exp(-t.^2); % Compute the Fourier transform F = fftshift(fft(f)) * dt; % Compute the inverse Fourier transform f_recovered = ifft(ifftshift(F)) / dt; % Analytical Fourier transform df = 1 / (N*dt); f_max = 1 / (2*dt); freq = linspace(-f_max, f_max, N); F_analytical = sqrt(pi) * exp(-(pi*freq).^2); % Plot results figure; % Original and recovered functions subplot(2,1,1); plot(t, f, 'b-', 'LineWidth', 2); hold on; plot(t, real(f_recovered), 'r--', 'LineWidth', 2); xlabel('Time'); ylabel('Amplitude'); title('Original and Recovered Gaussian Function'); legend('Original', 'Recovered (IFFT)'); grid on; % Fourier transform subplot(2,1,2); plot(freq, abs(F), 'b-', 'LineWidth', 2); hold on; plot(freq, F_analytical, 'r--', 'LineWidth', 2); xlabel('Frequency'); ylabel('Magnitude'); title('Fourier Transform of Gaussian Function'); legend('Numerical (FFT)', 'Analytical'); grid on; % Compare original and recovered functions max_diff = max(abs(f - real(f_recovered))); fprintf('Maximum difference between original and recovered: %.2e\n', max_diff);

                                                 *     *     *

% Define parameters N = 1000; % Number of points t_max = 5; % Maximum time value dt = 2*t_max / (N-1); t = linspace(-t_max, t_max, N); % Define the Gaussian function f = exp(-t.^2); % Analytical Fourier Transform df = 1 / (N*dt); f_max = 1 / (2*dt); freq = linspace(-f_max, f_max, N); F_analytical = sqrt(pi) * exp(-(pi*freq).^2); % Numerical Fourier Transform using FFT F_numerical = fftshift(fft(f)) * dt; % Plot results figure; plot(freq, abs(F_numerical), 'b-', 'LineWidth', 2); hold on; plot(freq, F_analytical, 'r--', 'LineWidth', 2); xlabel('Frequency'); ylabel('Magnitude'); title('Fourier Transform of Gaussian Function'); legend('Numerical (FFT)', 'Analytical');

grid on; % Compare numerical and analytical results max_diff = max(abs(abs(F_numerical) - F_analytical)); fprintf('Maximum difference between numerical and analytical: %.2e\n', max_diff);

It calculated a difference of .239 between the analytic Transform (the answer I provided)

and its own calculation of the Transform form the original function...