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A piecewise function is a type of function characterized by different expressions, depending on the input value's interval. These are particularly useful when a single formula cannot describe all cases of a situation. In this scenario, the function is defined within two distinct intervals for the domain

• For \(-\pi \leq x < 0\), the function value is 0.
• For \(0 \leq x \leq \pi\), the function value is 1.

This makes it a "piecewise function," because it stitches together different "pieces" of the function that apply within those specified domains. Understanding piecewise functions is critical for dealing with complex practical or theoretical problems that involve different conditions across various intervals.
Additionally, this piecewise behavior shows that our function is periodic with a period of \(2\pi\). Periodicity is important because it allows the use of Fourier Series to decompose or analyze the function.

Fourier coefficients are essential for transforming a function into a series of sines and cosines. These coefficients – specifically \(a_0\), \(a_k\), and \(b_k\) – help in constructing a Fourier series that can express the original function. The Fourier coefficients are significant for converting complex mathematical functions into simpler trigonometric terms.

• Average Value \(a_0\): This term represents the average value of the function over a period and is calculated using \(a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx\). It gives an offset for the series and forms the basis.
• Cosine Coefficients \(a_k\): For symmetry with cosine functions, these are found by \(a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(kx) \, dx\). These coefficients reflect how much of each cosine wave is needed.
• Sine Coefficients \(b_k\): Analogous to the cosine coefficients, sine coefficients are calculated by \(b_k = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(kx) \, dx\). These coefficients represent the sine wave components of the function.

In this exercise, \(a_k\) goes to 0 because the resulting cosine integral evaluates to zero for any integer \(k\), while \(b_k\) only results in values when \(k\) is odd, showing the exclusivity of sine components in this example.

Trigonometric integrals are a crucial part of finding Fourier coefficients, as they involve integrating functions multiplied by sine or cosine over a period. Here, each integral corresponds to assessing how much each trigonometric function resembles the target function over one complete cycle.

• Integrating Cosines: To find \(a_k\), you use \(\int_{0}^{\pi} \cos(kx) \, dx\). This integral focuses on cosine's contribution to reconstruction. In this scenario, it ultimately vanishes because \(\sin(k\pi) = 0\).
• Integrating Sines: For sine coefficients \(b_k\), you evaluate \(\int_{0}^{\pi} \sin(kx) \, dx\). This highlights how sine waves describe the original function's shape. It leads to non-zero outcomes, symbolizing the required sine components.

Applying these integrals effectively transforms the original piecewise function into its corresponding Fourier Series. This process helps in analyzing the function's behavior through its harmonic sine and cosine functions. The simplification seen in this exercise underscores the computed series' reflection of the piecewise function's periodic nature.