Fourier-Sine Series Expansion Of F(x) = X² - 4x On (0, 4)
Introduction to Fourier-Sine Series
In the realm of mathematical analysis, Fourier series play a pivotal role in representing periodic functions as an infinite sum of sines and cosines. This representation allows us to decompose complex functions into simpler, more manageable components, making it a powerful tool in various fields such as signal processing, heat transfer, and quantum mechanics. When dealing with functions defined on a finite interval, we can utilize either the Fourier-Sine or Fourier-Cosine series, depending on the boundary conditions and the desired symmetry of the representation. The Fourier-Sine series, in particular, is employed for functions that are zero at the endpoints of the interval, effectively creating an odd extension of the function. This article delves into the intricacies of expanding the function f(x) = x² - 4x over the interval (0, 4) into its Fourier-Sine series representation. We will explore the theoretical underpinnings of Fourier-Sine series, the computational steps involved in determining the coefficients, and the practical implications of this expansion. Understanding Fourier-Sine series is crucial for anyone working with periodic phenomena or boundary value problems, as it provides a systematic way to analyze and synthesize complex functions from their sinusoidal components. Furthermore, the application of Fourier-Sine series extends beyond theoretical mathematics, finding practical use in engineering and physics where the decomposition of signals and waveforms is essential for analysis and manipulation.
Problem Statement: Expanding f(x) = x² - 4x
Our primary objective is to expand the function f(x) = x² - 4x into a Fourier-Sine series over the interval (0, 4). This function, a simple quadratic, exhibits interesting behavior within the given interval, crossing the x-axis and changing sign. To achieve this expansion, we need to determine the Fourier-Sine coefficients, which dictate the amplitude of each sine term in the series. The Fourier-Sine series representation of a function f(x) defined on the interval (0, L) is given by:
f(x) = Σ[bₙ * sin(nπx/L)] (from n=1 to ∞)
where the coefficients bₙ are calculated using the integral:
bₙ = (2/L) ∫[f(x) * sin(nπx/L) dx] (from 0 to L)
In our case, L = 4, and f(x) = x² - 4x. The challenge lies in evaluating the integral for bₙ, which involves integrating the product of a quadratic function and a sine function. This requires the application of integration by parts, often multiple times, to systematically reduce the complexity of the integral. The resulting Fourier-Sine series will provide a representation of f(x) as an infinite sum of sine waves, each with a specific frequency and amplitude determined by the coefficients bₙ. This expansion is particularly useful for analyzing the function's behavior and for applications where sinusoidal components are of interest. The convergence of the series and the accuracy of the representation will also be important considerations as we proceed with the calculation of the coefficients.
Calculating the Fourier-Sine Coefficients
To determine the Fourier-Sine series expansion of f(x) = x² - 4x over the interval (0, 4), the crucial step is to calculate the coefficients bₙ. These coefficients quantify the contribution of each sine term in the series and are defined by the integral:
bₙ = (2/4) ∫[(x² - 4x) * sin(nπx/4) dx] (from 0 to 4)
This integral requires careful evaluation, as it involves the product of a quadratic function and a sine function. The standard technique for such integrals is integration by parts, which we will need to apply multiple times. Let's break down the process step by step. First, we set up the initial integration by parts:
Let:
- u = x² - 4x
- dv = sin(nπx/4) dx
Then:
- du = (2x - 4) dx
- v = - (4/nπ) cos(nπx/4)
Applying integration by parts, ∫u dv = uv - ∫v du, we get:
∫[(x² - 4x) * sin(nπx/4) dx] = - (4/nπ) (x² - 4x) cos(nπx/4) + (4/nπ) ∫[(2x - 4) cos(nπx/4) dx]
Now, we need to evaluate the new integral ∫[(2x - 4) cos(nπx/4) dx]. We apply integration by parts again:
Let:
- u = 2x - 4
- dv = cos(nπx/4) dx
Then:
- du = 2 dx
- v = (4/nπ) sin(nπx/4)
Applying integration by parts again, we get:
∫[(2x - 4) cos(nπx/4) dx] = (8/nπ) sin(nπx/4) + (32/(n²π²)) cos(nπx/4) - (32/(n²π²))
Substituting this back into the original expression and evaluating the definite integral from 0 to 4, we obtain a formula for bₙ. After simplifying, the expression for bₙ can be reduced significantly. This detailed calculation underscores the importance of methodical integration techniques and careful algebraic manipulation in determining Fourier-Sine coefficients. The resulting coefficients will allow us to construct the Fourier-Sine series representation of f(x).
Resulting Fourier-Sine Series
After performing the integration and simplification steps, the Fourier-Sine coefficients bₙ for the function f(x) = x² - 4x over the interval (0, 4) are found to be:
bₙ = (32/n³π³) * (1 - cos(nπ))
This expression for bₙ reveals an interesting pattern. When n is even, cos(nπ) = 1, and thus bₙ = 0. This means that only the odd terms contribute to the Fourier-Sine series. When n is odd, cos(nπ) = -1, and bₙ simplifies to 64/(n³π³). Therefore, the Fourier-Sine series for f(x) = x² - 4x over (0, 4) can be written as:
f(x) = Σ[(64/(n³π³)) * sin(nπx/4)] (from n=1, 3, 5, ... to ∞)
This series represents f(x) as an infinite sum of sine waves with frequencies that are odd multiples of the fundamental frequency π/4. The amplitude of each sine wave is determined by the coefficient 64/(n³π³), which decreases rapidly as n increases. This rapid decrease in amplitude indicates that the series converges relatively quickly, and a good approximation of f(x) can be obtained by summing only the first few terms of the series. The resulting Fourier-Sine series provides a powerful tool for analyzing and manipulating the function f(x). It allows us to decompose the function into its sinusoidal components, which can be useful in various applications, such as signal processing and the solution of differential equations. Furthermore, the series highlights the odd symmetry of the sine terms and their ability to represent the given function over the specified interval.
Convergence and Implications
The convergence of the Fourier-Sine series is a crucial aspect to consider when using it to represent a function. For the series we derived, f(x) = Σ[(64/(n³π³)) * sin(nπx/4)] (from n=1, 3, 5, ... to ∞), the coefficients bₙ decrease proportionally to 1/n³. This rapid decay in the coefficients ensures that the series converges pointwise to f(x) for 0 < x < 4. However, it's important to note that at the endpoints, x = 0 and x = 4, the Fourier-Sine series converges to 0, which is consistent with the odd periodic extension of f(x). The Gibbs phenomenon, a common characteristic of Fourier series, might be observed near points of discontinuity if the function were discontinuous, but in this case, f(x) = x² - 4x is continuous on the interval (0, 4), so the convergence is smooth. The implications of this Fourier-Sine series representation are far-reaching. It allows us to express a seemingly complex function as a sum of simple sinusoidal components, which are much easier to analyze and manipulate. In signal processing, this decomposition is invaluable for filtering, noise reduction, and spectral analysis. In the context of differential equations, the Fourier-Sine series can be used to solve boundary value problems, where the boundary conditions often dictate the use of sine or cosine series. Furthermore, the series provides insights into the frequency content of the function, revealing the dominant frequencies and their amplitudes. This information is crucial in understanding the behavior of the function and its response to various inputs or disturbances. The Fourier-Sine series also serves as a cornerstone for more advanced techniques, such as the Fourier transform, which extends the concept of frequency domain analysis to non-periodic functions. In summary, the convergence properties and the practical implications of the Fourier-Sine series highlight its significance as a fundamental tool in mathematics, physics, and engineering.
Conclusion
In conclusion, we have successfully expanded the function f(x) = x² - 4x into its Fourier-Sine series representation over the interval (0, 4). This involved calculating the Fourier-Sine coefficients, which were found to be bₙ = (32/n³π³) * (1 - cos(nπ)). We observed that only the odd terms contribute to the series, leading to the representation:
f(x) = Σ[(64/(n³π³)) * sin(nπx/4)] (from n=1, 3, 5, ... to ∞)
This Fourier-Sine series provides a powerful tool for analyzing and manipulating the function f(x). The rapid convergence of the series, due to the 1/n³ decay of the coefficients, ensures that a good approximation can be obtained by summing only the first few terms. The resulting sinusoidal decomposition reveals the frequency content of the function and allows for various applications in signal processing, boundary value problems, and other areas. The Fourier-Sine series is a testament to the power of representing functions as a sum of simpler components. By understanding the underlying principles and techniques involved in Fourier analysis, we gain valuable insights into the behavior of functions and their applications in diverse fields. This exploration of the Fourier-Sine series expansion of f(x) = x² - 4x not only demonstrates a specific mathematical procedure but also highlights the broader significance of Fourier analysis in mathematics, science, and engineering. The ability to decompose complex functions into simpler sinusoidal components is a cornerstone of modern analytical techniques, providing a framework for understanding and manipulating a wide range of phenomena.
