Question

Find the indicated Fourier series. Then differentiate your result repeatedly (both the function and the series) until you get a discontinuous function. Use a computer to plot $f(x)$ and the derivative functions. For each graph, plot on the same axes one or more terms of the corresponding Fourier series. Note the number of terms needed for a good fit (see comment at the end of the section). $$f(x)=\left(x^{2}-\pi^{2}\right)^{2}, \quad-\pi

Short answer

Find Fourier coefficients, construct the series, differentiate both repeatedly till discontinuity, plot on a computer, compare plots.

Step-by-step solution

Define the function and find the Fourier series

Given function is $f(x)=(x^2-{\pi }^2{)}^2$ over the interval $-\pi <x<\pi$. To find the Fourier series, compute the Fourier coefficients, $a_0$, $a_n$, and $b_n$ using the formulas:$$a_0=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)dx$$$$a_n=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\cos (nx)dx$$$$b_n=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\sin (nx)dx$$

Compute the coefficients

Calculate the integrals for $a_0$, $a_n$, and $b_n$. Evaluate these integrals step by step.$$a_0={\int }_{-\pi }^{\pi }(x^2-{\pi }^2{)}^{2}dx$$ and so on for $a_n$ and $b_n$. Note that $b_n$ will be zero due to the even nature of the function.

Write down the Fourier series

Construct the Fourier series using the coefficients found in Step 2. The general form is:$$f(x)=\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_n\cos (nx)+b_n\sin (nx))$$

Differentiate the Fourier series and the function

Differentiate both the function $f(x)$ and its Fourier series repeatedly until the result becomes discontinuous. Check each derivative step and verify the points of continuity.

Use a computer to plot the functions

Plot the original function $f(x)$ and its derivatives. For each graph, plot one or more terms of the Fourier series. Use software or programming languages like MATLAB, Python, or others to plot these accurately.

Analyze the plots

Compare the plots of the function and its Fourier series. Note the number of terms required to get a good fit for the function and its derivatives.

Key concepts

Fourier coefficients

Fourier coefficients are essential in decomposing a function into its sine and cosine components. To find these coefficients, we use integrals over one period of the function. The Fourier series of a function is expressed in terms of three types of coefficients: a₀, a_n, and b_n.

To compute the coefficients effectively:

• Find the average value of the function with a₀.
• Determine the weights of the cosine terms with a_n.
• Calculate the weights of the sine terms using b_n.

Because the given function is even, all sine terms (b_n) will be zero. Only cosine terms will contribute to the Fourier series representation of the function.

Remember, accurate computation of these integrals is crucial to having an accurate series representation.

Function differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change. Differentiating a Fourier series means we find the derivatives of the individual sine and cosine terms.

This process step-by-step involves:

• Applying standard differentiation rules to the Fourier series terms.
• Remembering to differentiate each term separately.
• Observing where the resulting derivative becomes discontinuous.

Since differentiation increases the frequency components' magnitude, higher points of discontinuity can appear, revealing more complex behavior in the function's behavior.

In practice, this helps us understand how the function changes and identifies the intricate details of its structure and growth.

Graphing software

Graphing software helps visualize functions and their Fourier series. It is crucial for comparing theoretical results with actual functions. Tools like MATLAB, Python (with libraries like Matplotlib or NumPy), and other plotting software make this task easier.

When using graphing software to plot Fourier series and their derivatives:

• Input the function and manually calculate the first few terms of its Fourier series.
• Use built-in functions to plot the original function and overlay Fourier series approximations.
• Differentiating both function and its series to overlay their plots and analyze results.
• Compare how well these series fit with increasing terms to gauge their accuracy.

These software tools facilitate a deeper understanding by providing visual and interactive insights into Fourier transformation and differentiation.

This approach confirms the mathematical results and enhances learning by visually demonstrating theoretical concepts in action.