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The half-range Fourier sine series is a powerful tool for representing a function over a given interval by expanding it using only sine terms. Sine functions are naturally odd, meaning that when we extend our original function, it becomes odd with respect to the y-axis. This allows the sine series to effectively express functions defined on half of a typical interval (such as 0 to \( \pi \)).

The sine series for a function \( f(t) \) is given by:

• \( f(t) \approx \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi t}{L}\right) \)
• where \( b_n = \frac{2}{L} \int_0^L f(t) \sin\left(\frac{n \pi t}{L}\right) \, dt \)

To find the coefficients \( b_n \), we integrate over the function\( f(t) \) multiplied by sine functions. Here, \( L = \pi \), which matches the upper limit of our given interval.

This method of using sine terms ensures that non-symmetrical parts in even functions are accurately represented by focusing solely on the periodic nature of odd extensions.

The half-range Fourier cosine series utilizes cosine terms to represent a function over a specified interval. Cosine functions are even, which plays an essential role when we extend a function, such as our given \( f(t) = \pi t - t^2 \), over a symmetric interval like \(-2\pi < t < 2\pi\).

The cosine series is expressed as:

• \( f(t) \approx \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi t}{L}\right) \)
• where \( a_0 = \frac{2}{L} \int_0^L f(t) \, dt \) and \( a_n = \frac{2}{L} \int_0^L f(t) \cos\left(\frac{n \pi t}{L}\right) \, dt \)

This involves finding the coefficients \( a_0 \) and \( a_n \) by integrating the function and its product with cosine terms. Here, \( L = \pi \).

The Fourier cosine series makes full use of the even extension of functions. As a result, it effectively maps the full shape of even expansions across the original half interval.

Integration by parts is a technique used to solve more complex integrals, and it's vital when calculating coefficients for Fourier series. Specifically, when dealing with products of polynomial and trigonometric functions, integration by parts can greatly simplify the process.

The formula for integration by parts is:

• \( \int u \, dv = uv - \int v \, du \)

In this formula, \( u \) and \( dv \) are parts of the original integral chosen such that \( du \) and \( v \) are simpler to work with.

For our function, \( f(t) = \pi t - t^2 \), we must compute both sine and cosine terms over integrations, especially when dealing with Fourier coefficients. Using proper selection for \( u \) and \( dv \), integration by parts helps in evaluating these integrals efficiently, ensuring that the terms contributing to the series coefficients are properly handled.

Periodic extension refers to extending a function, initially defined on a finite interval, so that it repeats its values in a cyclical pattern across an infinite domain. This is the foundation of Fourier series as it converts non-periodic functions into periodic ones.

In the context of half-range Fourier series, periodic extension is performed differently for sine and cosine series:

• Sine Series: Extend the function by mirroring and repeating it as an odd function.
• Cosine Series: Mirror and repeat it as an even function to maintain even symmetry.

For the given function \( f(t) = \pi t - t^2 \), extending it from 0 to \( \pi \) to span an interval \(-2\pi < t < 2\pi\) ensures it becomes part of a repeating pattern.

Such periodic extension not only allows the Fourier series to represent the function accurately outside its initial bounds but also provides the capability to analyze and understand its behavior across larger intervals seamlessly.