# Solving Fourier Problems: A Primer on the Piecewise Function f(x)

• Mindscrape
In summary, the conversation was about solving Fourier problems and the use of integration to find the "a" and "b" terms. The function discussed was a sawtooth and the Fourier series for it was given. Dirichlet's theorem was also mentioned as a condition for convergence of the series. The conversation ended with a question about the convergence of a particular series and the explanation that the limit of the generating function determines the convergence of its associated Fourier series.
Mindscrape
I was going through trying to solve various Fourier problems, and I came across this one.

$$f(x) = \left\{\begin{array} {c}0 \ \ \mbox{for} \ \ - \pi <x<0 & x \ \ \mbox{for} \ \ 0<x<\pi$$

Here is how far I have gotten, using that
$$a_n = \frac{1}{\pi} \int_{- \pi}^{\pi} xcosnx dx$$
and
$$b_n = \frac{1}{\pi} \int_{- \pi}^{\pi} xsinnx dx$$
arriving at

$$f(x) = \frac{\pi}{4} + (sinx - \frac{sin2x}{2} + \frac{sin3x}{3}- ...) + (unknown)$$

So, I figured out the "b" terms, and the initial "a" term was easy, but here is where I have a few doubts; hence, the "unknown." For values when the b term is not zero, I found the integration by parts to be

$$\frac{1}{\pi}([\frac{x}{n}sinnx]^{\pi}_{0} + \frac{1}{n}\int^{\pi}_{0}sinnx dx)$$ (1)

I know that the integral will come out to give

$$2(cosx+\frac{cos3x}{3^2} + \frac{cos5x}{5^2} + ...)$$

and the first term in the parts equation (1) will always be zero

So, then I think the answer will be

$$f(x) = \frac{\pi}{4} + (sinx - \frac{sin2x}{2} + \frac{sin3x}{3}- ...) + \frac{2}{\pi}(cosx+\frac{cos3x}{3^2} + \frac{cos5x}{5^2} + ...)$$

But, it just seems a little strange that a_n terms go by squares. Fourier analysis is new to me, and so I can't really gauge how certain functions look like.

Also, the other thing I wanted to double check on was how the function looked; this is a sawtooth, right?

P.S.
Does anybody know the piecewise function command for LaTeX? I thought it was \cases.
P.S.S
Thanks, hopefully the formatting is better now.

Last edited:
u mean like this?

$$|ax|=\left\{ \begin{array} {c} -ax \ \ \mbox{for} \ \ ax<0 & +ax \ \ \mbox{for} \ \ ax\geq 0 \end{array}$$

Last edited:
Another thing I am confused about is Dirichlet's theorem. An example I was looking at said that

$$f(x) = \frac{1}{4} + \frac{1}{\pi}(\frac{cosx}{1} - \frac{cos3x}{3} + \frac{cos5x}{5} ...) + \frac{1}{\pi}(\frac{sinx}{1} - \frac{2sin2x}{2} + \frac{sin3x}{3} + \frac{sin5x}{5} - \frac{2sin6x}{6} ...)$$
converges to 1/2 at x=0.

I can't see how the equation converges to that value. The Fourier series is some sort of pulse train, right? I have a guess, but it doesn't much sense in general. Approaching zero from the left the value will be one, and approaching from the right it will be zero, so take the average?

Dirichlet's thm gives a weak conditon for convergence and says that when the series converges, it does so to the average value of the limit of the "generating function"* from the right and the limit from the left. This is useful when you want to know to which value does the Fourier series converges at points where there is a pointwise discontinuity.

*the function that we're considering the Fourier series of.

the moral of the story is that you only have to look at the generating function to know what its associated Fourier series converges to. you were looking at the series itself.

## 1. What is a Fourier problem?

A Fourier problem involves finding the coefficients of a Fourier series for a given function. This allows for the representation of a function as a sum of sines and cosines, which can be useful in analyzing and solving differential equations.

## 2. What is a piecewise function?

A piecewise function is a function that is defined by different rules or equations for different parts of its domain. This means that the function may have a different expression or behavior depending on which interval the input falls into.

## 3. How do you solve a Fourier problem?

To solve a Fourier problem, you first need to determine the interval of the piecewise function and the coefficients of the sine and cosine terms for each interval. This can be done by using the Fourier series formula and integrating the given function over the interval. Then, you can plug in the coefficients and simplify the expression to get the final solution.

## 4. What are some real-world applications of solving Fourier problems?

Solving Fourier problems has many practical applications in fields such as engineering, physics, and signal processing. It can be used to model and analyze physical systems, design filters, and compress data. It is also useful in solving differential equations that arise in various scientific and engineering problems.

## 5. Are there any limitations to using Fourier series to solve problems?

While Fourier series can be a powerful tool for solving problems, there are some limitations. It may not always be possible to represent a function accurately using a finite number of sine and cosine terms. Additionally, some functions may have discontinuities or singularities that make it difficult to find a suitable Fourier series representation. In these cases, other methods may need to be used.

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