Fourier series and the dirchlet integral

• dirk_mec1
In summary, the conversation discusses using a Fourier series to prove that the integral of sin(x)/x from 0 to infinity equals pi/2. The participants also mention trying to Fourier-transform sin(x) and sin(x)/x, but ultimately finding that it may not provide an easy solution.
dirk_mec1
Use a Fourier series to prove that: $$\int_{0}^{ \infty} \frac{\sin(x)}{x}\ \mbox{d}x = \frac{ \pi}{2}$$

What function do I need to Fourier transform?

There aren't many to choose from are they? I suggest you try Fourier-transforming sin(x) and sin(x)/x and see where it leads you.

Pere Callahan said:
There aren't many to choose from are they? I suggest you try Fourier-transforming sin(x) and sin(x)/x and see where it leads you.

Actually there are and I found that there are two ways of finding this answer both NOT with solely a sin(x) or sin(x)/x. I'm suprised that you didn't took the effort to look more closely to the question at hand.

dirk_mec1 said:
Actually there are and I found that there are two ways of finding this answer both NOT with solely a sin(x) or sin(x)/x. I'm suprised that you didn't took the effort to look more closely to the question at hand.

You're right. In particular I read Fouriertransform instead of Fourier Series. You're also right in saying that transforming sin(x) or sin(x)/x seems not to provide an easy solution to the problem.

Which two ways are you referring to? One is likely to Laplace-transform 1/x ..?

1. What is a Fourier series and how is it used in mathematics?

A Fourier series is a mathematical representation of a periodic function as a sum of simple sine and cosine functions. It was developed by Joseph Fourier in the early 19th century and is used in various areas of mathematics, such as signal processing, differential equations, and harmonic analysis.

2. What is the difference between a Fourier series and a Fourier transform?

A Fourier series is used to represent a periodic function, while a Fourier transform is used to represent a non-periodic function. The Fourier transform also uses a continuous spectrum of frequencies, while the Fourier series uses a discrete set of frequencies.

3. What is the Dirichlet integral and why is it important in Fourier series?

The Dirichlet integral is a type of improper integral that is used to determine the convergence of a Fourier series. It is important because it helps to determine whether a given function can be represented by a Fourier series and if so, in what form.

4. How is the Dirichlet integral calculated?

The Dirichlet integral is calculated by taking the limit of the integral of a function over a specific interval as the interval approaches infinity. This is known as an improper integral and can be evaluated using various techniques such as the comparison test or the Cauchy criterion.

5. What are some applications of Fourier series and the Dirichlet integral?

Fourier series and the Dirichlet integral have numerous applications in mathematics and engineering. They are used in signal processing to analyze and filter signals, in image processing to compress images, in solving differential equations, and in solving boundary value problems in physics and engineering. They are also used in the study of heat transfer, fluid dynamics, and acoustics.

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