Fourier Series Of F(x) = |x|: A Step-by-Step Guide

Let's dive into finding the Fourier series for the even function f(x) = |x|. This is a classic problem in Fourier analysis, and understanding it will give you a solid foundation for tackling more complex series. We'll break it down step-by-step, making sure everything is clear and easy to follow.

Understanding Even Functions and Fourier Series

Before we jump into the math, let's quickly recap what even functions and Fourier series are. This will help solidify our understanding and make the process smoother.

• Even Function: A function f(x) is even if f(-x) = f(x) for all x in its domain. Geometrically, this means the graph of the function is symmetric about the y-axis. The absolute value function, f(x) = |x|, perfectly fits this description. Think about it: |2| = 2 and |-2| = 2. Because of this symmetry, the Fourier series of an even function simplifies considerably; it contains only cosine terms. This is a huge advantage! We don't have to worry about calculating sine coefficients. Remember this, it's key!

• Fourier Series: A Fourier series is a way to represent a periodic function as an infinite sum of sines and cosines. The general form of a Fourier series for a function f(x) defined on the interval [-L, L] is:

  f(x) = a₀/2 + Σ[ₙ=₁ to ∞] (aₙ cos(nπx/L) + bₙ sin(nπx/L)),

  where a₀, aₙ, and bₙ are the Fourier coefficients. These coefficients determine the amplitude of each sine and cosine term in the series. Finding these coefficients is the main task when determining the Fourier series of a function. This decomposition into sines and cosines is incredibly powerful, allowing us to analyze and manipulate complex functions in terms of simpler, more manageable components. The values of a₀, aₙ, and bₙ are calculated using integrals that capture the correlation between the original function and the respective sine and cosine functions. The interval [-L, L] defines the period over which the function is being analyzed; outside of this interval, the function is assumed to repeat periodically.

Because f(x) = |x| is even, all the bₙ coefficients will be zero. This significantly simplifies our calculations. We only need to focus on finding a₀ and the aₙ coefficients. This property stems directly from the orthogonality of sine and cosine functions over a symmetric interval when multiplied by an even function. The integral of an odd function over a symmetric interval is always zero, and since sine functions are odd, the product of an even function and a sine function will also be odd. Thus, the integral that defines the bₙ coefficients vanishes, leaving us only with the cosine terms to consider.

Defining the Function and Interval

Okay, so we're looking at f(x) = |x|. To define our problem fully, we need to specify the interval over which we want to represent this function as a Fourier series. Let's choose the interval [-π, π]. This is a common and convenient choice, but remember, you can choose other intervals as needed. The choice of interval affects the values of the Fourier coefficients and the convergence of the series. A different interval would simply rescale the cosine terms and potentially change the rate at which the series converges to the function. We need to be mindful of these interval boundaries when calculating the integrals for the coefficients.

So now we have:

• f(x) = |x|
• Interval: [-π, π]
• L = π (half the length of the interval)

This means our Fourier series will look like:

f(x) = a₀/2 + Σ[ₙ=₁ to ∞] aₙ cos(nx). Notice the π in the cosine argument has canceled out since L = π. This simplification is a direct result of choosing the interval [-π, π]. If we had chosen a different interval, say [-L, L], the cosine argument would be nπx/L, and the calculations would be slightly more involved. Always pay close attention to the interval when setting up your Fourier series.

Calculating the Fourier Coefficients

Now for the fun part – calculating those Fourier coefficients! We'll start with a₀ and then move on to the aₙ coefficients.

Calculating a₀

The formula for a₀ is:

a₀ = (1/L) ∫[-ₗ to ₗ] f(x) dx

In our case, L = π and f(x) = |x|, so:

a₀ = (1/π) ∫[-π to π] |x| dx

Since |x| is an even function, we can simplify this integral:

a₀ = (2/π) ∫[₀ to π] x dx

Now, integrate:

a₀ = (2/π) [x²/2] from 0 to π

a₀ = (2/π) (π²/2 - 0)

a₀ = π

So, a₀ = π. This is the average value of the function |x| over the interval [-π, π]. The factor of 1/π scales the integral to represent the average, and the result, π, gives us the DC component of the Fourier series. This value will contribute a constant term to our series representation of f(x).

Calculating aₙ

The formula for aₙ is:

aₙ = (1/L) ∫[-ₗ to ₗ] f(x) cos(nπx/L) dx

Again, L = π and f(x) = |x|, so:

aₙ = (1/π) ∫[-π to π] |x| cos(nx) dx

Since both |x| and cos(nx) are even functions, their product is also even. Therefore, we can simplify the integral:

aₙ = (2/π) ∫[₀ to π] x cos(nx) dx

Now, we need to use integration by parts. Let:

u = x and dv = cos(nx) dx

Then:

du = dx and v = (1/n) sin(nx)

Using the integration by parts formula, ∫u dv = uv - ∫v du:

aₙ = (2/π) [ (x/n) sin(nx) | from 0 to π - ∫[₀ to π] (1/n) sin(nx) dx ]

Let's evaluate the first term:

(x/n) sin(nx) | from 0 to π = (π/n) sin(nπ) - (0/n) sin(0) = 0

Since sin(nπ) = 0 for all integers n. Now, let's evaluate the remaining integral:

- ∫[₀ to π] (1/n) sin(nx) dx = (1/n²) cos(nx) | from 0 to π = (1/n²) [cos(nπ) - cos(0)]

Recall that cos(nπ) = (-1)ⁿ and cos(0) = 1, so:

(1/n²) [(-1)ⁿ - 1]

Putting it all together:

aₙ = (2/π) * (1/n²) [(-1)ⁿ - 1]

Now, let's analyze this result. If n is even, then (-1)ⁿ = 1, so aₙ = 0. If n is odd, then (-1)ⁿ = -1, so aₙ = (2/π) * (1/n²) * (-2) = -4/(πn²). Therefore,

aₙ = { -4/(πn²) if n is odd, 0 if n is even }

Constructing the Fourier Series

Now that we have a₀ and aₙ, we can construct the Fourier series for f(x) = |x| on the interval [-π, π]. Recall that:

f(x) = a₀/2 + Σ[ₙ=₁ to ∞] aₙ cos(nx)

Substituting our values:

f(x) = π/2 + Σ[ₙ=₁ to ∞] aₙ cos(nx)

Since aₙ = 0 for even n, we only need to sum over odd n. Let n = 2k - 1, where k goes from 1 to infinity. Then:

f(x) = π/2 + Σ[ₖ=₁ to ∞] (-4/(π(2k-1)²)) cos((2k-1)x)

So, the Fourier series for f(x) = |x| on the interval [-π, π] is:

f(x) = π/2 - (4/π) Σ[ₖ=₁ to ∞] (1/(2k-1)²) cos((2k-1)x)

This series converges to |x| for x in the interval [-π, π]. Outside of this interval, the series represents the periodic extension of |x| with period 2π. The convergence is generally good, but it's worth noting that Fourier series can exhibit the Gibbs phenomenon at points of discontinuity (though |x| itself is continuous, its derivative isn't at x=0).

Verification and Visualization

To verify our result, we can plot the partial sums of the Fourier series and compare them to the original function, f(x) = |x|. As we include more terms in the series, the approximation should become more accurate. Software like Python with libraries such as NumPy and Matplotlib makes this verification process straightforward. You can plot both the original function and the truncated Fourier series to visually assess the accuracy of the approximation. Specifically, you'd want to observe how the partial sums converge to |x| within the interval [-π, π], paying attention to any overshoot or ringing near the point x = 0, where the derivative is discontinuous. Furthermore, you can calculate the mean squared error between the partial sum and the original function to quantify the convergence as you increase the number of terms. This numerical analysis provides strong evidence for the correctness of the derived Fourier series.

This step is crucial to confirm our calculations and gain a deeper understanding of how the Fourier series represents the function. It also helps to illustrate the power and limitations of Fourier series approximations.

Conclusion

Finding the Fourier series of f(x) = |x| involves understanding even functions, applying the Fourier series formulas, and carefully calculating the Fourier coefficients. The result is a series that represents |x| as an infinite sum of cosine functions. Remember, the key takeaways are: 1) identify even/odd symmetry to simplify calculations, 2) meticulously apply integration by parts, and 3) carefully evaluate the resulting expressions, paying close attention to boundary conditions and trigonometric identities. And guys, don't forget to verify your results visually or numerically! This not only confirms the accuracy of your calculations but also provides valuable insight into the behavior of Fourier series. Now you're equipped to tackle similar problems with confidence! Keep practicing, and you'll become a Fourier series master in no time!