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Figure 8.P.3 Overview

• Left Column: Different functions of time. Each plot maintains the same vertical and horizontal scale for comparison.

• Right Column: Magnitudes of the Fourier amplitudes for these functions, though the order is scrambled.

• Definitions:

  • Fourier Amplitude (An): Represents the amplitude in the Fourier expansion of a function. This is crucial for analyzing the frequency components of the time-dependent functions.

Functions of Time (Left Column)

• Each panel in the left column depicts a unique time-dependent function (labeled a, b, c, d, e, f, g, h, i, j, k).

• The analysis requires pairing these functions with the corresponding Fourier amplitudes displayed in the right column.

Fourier Amplitudes (Right Column)

• The right column displays the Fourier amplitudes plotted as a function of frequency (w), maintaining the same horizontal scale for comparison across different functions.

• Magnified Insets: Panels j and k provide detailed views of the low-frequency region, emphasizing variations in amplitude at lower frequencies.

Specific Observations

• Panels h and i: Each features only a single dot in the top left corner, indicating that they have one non-zero Fourier amplitude. This is significant for understanding the function's frequency representation.

Analysis Requirement

• Matching Pairs: Identify and explain which Fourier amplitude corresponds to each time function in the left column. This requires an understanding of how each function's characteristics influence its Fourier representation.