Question

Sketch (by hand) the graphs of each of the following functions over four periods. Then sketch the extensions of each of the functions as both an even and odd periodic function. Determine the corresponding Fourier sine and cosine series, and verify the convergence to the desired function using Maple. a. \(f(x)=x^{2}, 0

Short answer

The problem involves understanding the properties of even and odd functions, creating their respective sketches over four periods and finding the Fourier sine and cosine series for each. The convergence to the original function for each case is then verified.

Step-by-step solution

Sketch Functions

First step is to sketch the given functions over four periods. That means repeating the same pattern of the function four times over. For instance, \(f(x) = x^2, 0<x<1\) is a simple quadratic function that goes from 0 to 1 within the interval. This pattern should be repeated four times over the x-axis in the graph.

Extend as Even and Odd Functions

The next step is to extend these functions as both even and odd by observing the symmetry of the function. An even function is symmetrical about the y-axis, meaning that for all x in the domain, \(f(x) = f(-x)\). On the other hand, an odd function has symmetry about the origin, which means \(f(x) = -f(-x)\) for all x in the domain. For instance, \(x^2\) is an even function.

Calculate Fourier Series

The next task is to find the Fourier sine and cosine series. Fourier series represent the function as a sum of sine and cosine functions that multiply together to generate the original function. The coefficients for the sine and cosine terms can be determined using the Fourier series formulas.

Verifying Convergence

The last step is to verify that each Fourier series converges to the original function by plotting them on the same graph. If the Fourier series is a good representation, it should coincide with the original function.

Key concepts

Periodic Functions

Periodic functions are a fundamental concept in mathematics where a function repeats its values in regular intervals or periods. A classic example is the sine and cosine functions, which repeat their pattern every 360 degrees or 2π radians.

When dealing with periodic functions, it's important to grasp that after a certain interval, known as the period, the function's values recur identically. This feature is pivotal in Fourier analysis, where periodic functions are expressed as a sum of sines and cosines.

In the given exercise, you're asked to sketch functions over four periods. This means you should take the basic pattern of the function over its initial interval and replicate it four times. This visualization helps in understanding how the function behaves over repeated intervals and lays the groundwork for extending them into Fourier series representations.

A key point about periodic functions is that they can be extended either as even or odd functions. Understanding this extension is crucial when working towards their Fourier representation.

Even and Odd Functions

Even and odd functions are categories based on symmetry. These functions are integral when forming Fourier series, as they dictate which components (sine or cosine) are present.

**Even Functions**: An even function is symmetrical about the y-axis. Mathematically, for any function \[ f(x) = f(-x) \]An example is the square function \(x^2\), which remains unchanged if you replace \(x\) with \(-x\). When analyzing Fourier series, even functions contribute only cosine terms.

**Odd Functions**: Odd functions, on the other hand, have rotational symmetry about the origin. That means, for any function:\[ f(x) = -f(-x) \]A simple example is \(x^3\). In Fourier expansions, odd functions contribute sine terms.

When extending periodic functions, identifying whether they are even or odd is essential as it directly impacts their Fourier series components. Understanding these symmetries can simplify analysis significantly.

Graph Sketching

Graph sketching is a vital skill for visualizing mathematical functions. It involves plotting the critical features of a function across several periods, especially when dealing with periodic functions.

Step-by-step sketching ensures a clear representation of the function's behavior. Here are some steps to follow:

• Identify the interval over which the function is initially defined.
• Determine the core features of the graph during this interval: peaks, troughs, intercepts.
• Replicate this pattern over the desired number of periods. For your task, extend it four times.

These steps help ensure you understand the periodic nature of the function and clearly see how it can be extended as even or odd.

Sketching also prepares you for the Fourier series extension, where understanding the symmetry (even or odd) can simplify the expansion process. By visualizing the function, you enhance comprehension and ensure accuracy in future analysis like determining Fourier coefficients.