Half-Range Fourier Cosine Series And Periodic Function Analysis

In this comprehensive article, we delve into two fundamental concepts in mathematical analysis: half-range Fourier cosine series and the analysis of periodic functions. These techniques are essential tools for representing and understanding various phenomena in physics, engineering, and other scientific disciplines. We will explore the process of determining the half-range Fourier cosine series for a given function and analyze a periodic function by sketching its graph and determining its Fourier series representation. This exploration will provide a solid foundation for further studies in signal processing, differential equations, and applied mathematics. This article aims to provide a detailed explanation with examples, making it accessible and informative for students, engineers, and anyone interested in the practical applications of Fourier analysis.

2.1 Understanding Half-Range Fourier Cosine Series

The half-range Fourier cosine series is a special type of Fourier series that represents a function defined on an interval (0, L) as an infinite sum of cosine terms. This series is particularly useful for representing functions that are even or can be extended as even functions. Unlike the full Fourier series, which includes both sine and cosine terms, the half-range cosine series consists only of cosine terms, making it simpler and more suitable for certain applications. The key idea behind using a cosine series is to extend the given function f(t) from the interval (0, L) to an even function on the interval (-L, L). This even extension ensures that the Fourier series will only contain cosine terms, as sine terms are odd functions and will have zero coefficients for an even function. In this section, we will systematically determine the half-range Fourier cosine series for the function f(t) = t, defined on the interval 0 < t ≤ π. This example will illustrate the general procedure and highlight the important steps involved in calculating the coefficients and constructing the series.

2.2 Calculating the Coefficients

To determine the half-range Fourier cosine series of the function f(t) = t on the interval 0 < t ≤ π, we need to calculate the cosine coefficients. The general form of the half-range Fourier cosine series is given by:

f(t) = A₀/2 + Σ[An * cos(nπt/L)]

where the summation is from n = 1 to ∞, L is the length of the interval, and A₀ and An are the coefficients. In our case, L = π, so the series becomes:

f(t) = A₀/2 + Σ[An * cos(nt)]

Now, we need to calculate the coefficients A₀ and An. The formula for A₀ is:

A₀ = (2/L) ∫[f(t) dt]

where the integral is taken from 0 to L. Plugging in f(t) = t and L = π, we get:

A₀ = (2/π) ∫[t dt] from 0 to π
   = (2/π) [t²/2] from 0 to π
   = (2/π) (π²/2)
   = π

So, A₀ = π. Next, we calculate the coefficients An using the formula:

An = (2/L) ∫[f(t) * cos(nπt/L) dt]

Again, with f(t) = t and L = π, we have:

An = (2/π) ∫[t * cos(nt) dt] from 0 to π

To solve this integral, we use integration by parts. Let u = t and dv = cos(nt) dt. Then, du = dt and v = (1/n)sin(nt). Applying integration by parts:

∫[t * cos(nt) dt] = t * (1/n)sin(nt) - ∫[(1/n)sin(nt) dt]
                 = (t/n)sin(nt) + (1/n²)cos(nt)

Now, we evaluate this from 0 to π:

[(π/n)sin(nπ) + (1/n²)cos(nπ)] - [0 + (1/n²)cos(0)]
= (π/n)(0) + (1/n²)cos(nπ) - (1/n²)(1)
= (1/n²)cos(nπ) - 1/n²
= (1/n²)[cos(nπ) - 1]

Thus,

An = (2/π) * (1/n²)[cos(nπ) - 1]

We know that cos(nπ) = (-1)^n, so

An = (2/(πn²))[(-1)^n - 1]

When n is even, (-1)^n = 1, so An = 0. When n is odd, (-1)^n = -1, so An = (2/(πn²))(-2) = -4/(πn²). Therefore, we can write An as:

An = {-4/(πn²) if n is odd; 0 if n is even}

2.3 Constructing the Fourier Series

Now that we have calculated A₀ and An, we can construct the half-range Fourier cosine series for f(t) = t on the interval 0 < t ≤ π. The series is:

f(t) = π/2 + Σ[An * cos(nt)]

Substituting the values of An, we get:

f(t) = π/2 + Σ[(-4/(πn²)) * cos(nt)]

where the summation is over odd values of n. We can write this out explicitly for the first few terms:

f(t) = π/2 - (4/π)cos(t) - (4/(9π))cos(3t) - (4/(25π))cos(5t) - ...

This series represents the function f(t) = t on the interval 0 < t ≤ π as a sum of cosine functions. The half-range Fourier cosine series is particularly useful because it allows us to approximate a function using a series of cosine terms, which are often easier to work with in applications such as signal processing and solving differential equations. By including more terms in the series, we can obtain a better approximation of the original function. This method is crucial for representing periodic signals and solving boundary value problems in various fields of science and engineering.

3.1 Defining the Periodic Function

In this section, we shift our focus to analyzing a periodic function. A periodic function is a function that repeats its values in regular intervals or periods. Understanding and representing periodic functions is crucial in many areas of science and engineering, such as signal processing, electrical engineering, and physics. The function we will analyze is defined as follows:

f(t) = { t² + t, for -2 < t < 2; f(t+4) otherwise }

This function is defined by a quadratic expression, t² + t, on the interval -2 < t < 2, and it is periodic with a period of 4. This means that the function repeats its pattern every 4 units along the t-axis. To fully understand this function, we need to sketch its graph and determine its Fourier series representation. Sketching the graph helps us visualize the function's behavior, while determining the Fourier series allows us to express the function as a sum of sine and cosine waves. This representation is particularly useful for analyzing the frequency components of the function and for solving differential equations involving periodic functions.

3.2 Sketching the Graph

To sketch the graph of the periodic function f(t), we first consider the interval -2 < t < 2, where f(t) = t² + t. This is a quadratic function, and its graph is a parabola. We can find the vertex of the parabola by completing the square or by using the formula t = -b/(2a) for the vertex of a quadratic function at² + bt + c. In this case, a = 1 and b = 1, so the t-coordinate of the vertex is t = -1/(21) = -0.5*. The value of the function at this point is f(-0.5) = (-0.5)² + (-0.5) = 0.25 - 0.5 = -0.25. Thus, the vertex of the parabola is at (-0.5, -0.25).

Now, we evaluate the function at the endpoints of the interval. At t = -2, f(-2) = (-2)² + (-2) = 4 - 2 = 2. At t = 2, f(2) = (2)² + (2) = 4 + 2 = 6. So, the parabola passes through the points (-2, 2) and (2, 6). We can now sketch the parabola on the interval -2 < t < 2. Since the function is periodic with a period of 4, we repeat this pattern for every interval of length 4. This means that the graph in the interval (2, 6) will be a shifted version of the graph in (-2, 2), and so on.

To sketch the graph for the interval 2 < t < 6, we note that f(t) = f(t - 4). So, the graph in this interval is the same as the graph in (-2, 2), but shifted 4 units to the right. Similarly, the graph in the interval (-6, -2) will be the same as the graph in (-2, 2), but shifted 4 units to the left. By repeating this pattern, we can sketch the graph of the periodic function f(t) over any interval. The resulting graph will consist of repeated segments of the parabola, each spanning an interval of length 4. This visual representation helps us understand the function's behavior and periodicity, which is crucial for further analysis and applications.

3.3 Determining the Fourier Series

To determine the Fourier series of the periodic function f(t), we need to calculate the Fourier coefficients. The general form of the Fourier series for a function f(t) with period T is:

f(t) = A₀/2 + Σ[An * cos(2πnt/T)] + Σ[Bn * sin(2πnt/T)]

where the summations are from n = 1 to ∞, and A₀, An, and Bn are the Fourier coefficients. In our case, the period T = 4, so the series becomes:

f(t) = A₀/2 + Σ[An * cos(πnt/2)] + Σ[Bn * sin(πnt/2)]

Now, we need to calculate the coefficients A₀, An, and Bn. The formula for A₀ is:

A₀ = (2/T) ∫[f(t) dt]

where the integral is taken over one period. In our case, we integrate from -2 to 2:

A₀ = (1/2) ∫[t² + t dt] from -2 to 2
   = (1/2) [t³/3 + t²/2] from -2 to 2
   = (1/2) [(8/3 + 2) - (-8/3 + 2)]
   = (1/2) (16/3)
   = 8/3

So, A₀ = 8/3. Next, we calculate the coefficients An using the formula:

An = (2/T) ∫[f(t) * cos(2πnt/T) dt]

With f(t) = t² + t and T = 4, we have:

An = (1/2) ∫[(t² + t) * cos(πnt/2) dt] from -2 to 2

We can split this integral into two parts:

An = (1/2) ∫[t² * cos(πnt/2) dt] from -2 to 2 + (1/2) ∫[t * cos(πnt/2) dt] from -2 to 2

The second integral is an integral of an odd function over a symmetric interval, so it is zero. The first integral can be solved using integration by parts twice. After performing the integration and evaluating the limits, we find:

An = (8/π²n²) * (-1)^n

Now, we calculate the coefficients Bn using the formula:

Bn = (2/T) ∫[f(t) * sin(2πnt/T) dt]

With f(t) = t² + t and T = 4, we have:

Bn = (1/2) ∫[(t² + t) * sin(πnt/2) dt] from -2 to 2

Again, we can split this integral into two parts:

Bn = (1/2) ∫[t² * sin(πnt/2) dt] from -2 to 2 + (1/2) ∫[t * sin(πnt/2) dt] from -2 to 2

The first integral is an integral of an even function times an odd function over a symmetric interval, so it is zero. The second integral can be solved using integration by parts. After performing the integration and evaluating the limits, we find:

Bn = (-4/(πn)) * (-1)^n

Having calculated A₀, An, and Bn, we can construct the Fourier series for f(t):

f(t) = 4/3 + Σ[(8/π²n²) * (-1)^n * cos(πnt/2)] + Σ[(-4/(πn)) * (-1)^n * sin(πnt/2)]

This series represents the periodic function f(t) as a sum of sine and cosine waves. The Fourier series is a powerful tool for analyzing periodic functions because it decomposes the function into its constituent frequencies. This representation is invaluable in fields such as signal processing, where understanding the frequency components of a signal is crucial for tasks like filtering and compression. By examining the coefficients of the series, we can determine the amplitude and phase of each frequency component, providing a comprehensive understanding of the function's behavior.

In this article, we have explored the concepts of half-range Fourier cosine series and the analysis of periodic functions. We systematically determined the half-range Fourier cosine series for f(t) = t on the interval 0 < t ≤ π, and we analyzed the periodic function f(t) = { t² + t, for -2 < t < 2; f(t+4) otherwise } by sketching its graph and determining its Fourier series representation. These techniques are fundamental in mathematical analysis and have wide-ranging applications in various scientific and engineering disciplines. Understanding these concepts provides a solid foundation for further studies in areas such as differential equations, signal processing, and applied mathematics. The ability to represent functions as Fourier series allows us to analyze their frequency components and solve complex problems involving periodic phenomena. By mastering these techniques, students, engineers, and scientists can gain valuable insights into the behavior of physical systems and develop effective solutions to real-world problems.