KFC said:In wiki (http://en.wikipedia.org/wiki/Fourier_transform), there said the Fourier transform of the Bessel function (zeroth order J0) is a rect function (window). But I also saw a text (about optics) that the Fourier transform on a ring slit (or ring disk) is zeroth-order Bessel function, so which one is correct? If wiki is correct, what is the Fourier transform on a ring disk?
The Fourier transformation of J0(x) is a mathematical process that converts a function from its original domain (often time or space) to its representation in the frequency domain. In other words, it decomposes the function into its constituent frequencies.
J0(x) is a special type of Bessel function, which is a class of mathematical functions that arise in various physical and engineering problems. In the Fourier transformation, J0(x) is often used as a basis function to represent signals or functions that have circular symmetry.
The Fourier transformation of J0(x) can be calculated using the integral representation of the Bessel function. This involves integrating the function over the entire real line and multiplying it by a complex exponential factor. The resulting expression is a series of complex numbers, known as the Fourier coefficients.
The Fourier transformation of J0(x) has numerous applications in various fields, including signal processing, image and audio compression, and solving differential equations. It is also used in Fourier optics, which studies the behavior of light waves and their interactions with optical systems.
Like any mathematical tool, the Fourier transformation of J0(x) has its limitations. It may not be suitable for all types of functions, and it has certain assumptions and constraints that must be met for it to be valid. Additionally, the calculation of the Fourier coefficients can be computationally intensive for complex functions.