What are the real world applications of the function sin(x)/x?

[JimfromRacine]

What use in real world calculation is frac{sin x} / x ?

 

[Hertz]

What use in real world calculation is frac{sin x} / x ?

One 'real world calculation' that you might do is take the derivative of that function in some problem that you might encounter if you are an engineer or scientist or something like that. The concept of $\sin{x}$ function and the $x$ function themselves lead to more real world applications than you could ever count, but the quotient $\sin{x}/x$ seems to be more interesting when viewed from a mathematical perspective.

 

[olivermsun]

What use in real world calculation is frac{sin x} / x ?

The sinc function: $\mathrm{sinc}(x) \equiv \dfrac{\sin x}{x}$ is the Fourier transform of a rectangular window or pulse. As a result, there are many real-world applications to signal processing. In particular, $\mathrm{sinc}(x)$ is the kernel for continuous reconstruction of a bandlimited, discrete-sampled signal (e.g., digital audio).

For not-unrelated reasons, it is also the characteristic pattern produced by single-slit diffraction.

 

[homeomorphic]

That sort of question might be a sign of taking the idea of "real-world" applicability a little too far. Sometimes, things can have a real-world "use" if they merely strengthen your understanding of real world phenomenon, rather than being directly applicable. It can sometimes be a piece of the fabric that holds a subject together.

In contrast, I got disillusioned with topology on the basis of entire books and hundreds and hundreds of pages that I could not see any real world application for, as opposed to one single function that can fit in one line, which might serve as a useful example to illustrate the theory. That should put it into perspective. I don't think it make sense to question the applicability of every little example. It's just something to practice on and gain experience with the subject, and that is real-world applicability. I'm not saying it's not a fair question to ask or an interesting question, but the way it is phrased suggests there might be some kind of underlying over-thinking going on.

 

[CellCoree]

The function sin(x)/x has numerous applications in real world calculations, particularly in the fields of mathematics and engineering. One example is in Fourier analysis, where the function is used to represent periodic signals and to analyze their frequency components. In signal processing, the function is used in the design of filters, which are essential for removing unwanted noise from signals.

In addition, the function is also used in physics, particularly in the study of wave phenomena. It is a fundamental component in the mathematical representation of diffraction patterns, which are observed when waves encounter obstacles or pass through small openings. This is crucial in understanding and predicting the behavior of light, sound, and other types of waves in various real world scenarios.

Moreover, the function has applications in finance, specifically in the calculation of interest rates and compound growth. It is also used in statistics, where it is utilized in probability distributions and in the analysis of data sets.

Overall, the real world use of sin(x)/x extends beyond just mathematical calculations and has practical applications in a wide range of fields. Its versatility and importance in various disciplines make it a fundamental function in scientific and technological advancements.