Taking foureir integral in polar coordinate

In summary: Fourier transform of the given function in the integral, and then evaluating it. In this case, the Fourier transform of $f(x,y)$ is given by:$$F(k_x,k_y) = \frac{2}{\lambda} \exp \left(-j \frac{4 \pi}{\lambda} \right) G(\theta).$$Substituting this expression in the integral, we get:$$\mathcal{F}[f(x,y)] = \int_0^{2 \pi} \int_0^{\infty} \frac{2}{\lambda} \exp \left(-j \frac{4 \pi}{\lambda} \rho \right) G(\theta) \rho
  • #1
h_cet
3
0
hi;

I have a question about taking the foureir trasnform in polar coordinate... the question is as following;
https://www.physicsforums.com/attachment.php?attachmentid=17892&stc=1&d=1236524155

I would like to learn that according to outermost integral ([tex]\rho[/tex]), the integral is taken. But here I would like to now how it is happened step by step... because in the second equation we have 2/[tex]\lambda[/tex] and exp(-j4pi 1/[tex]\lambda[/tex]) and G([tex]\theta[/tex])... How were they put in the equation?...

if you can explain it, I will be really glad...

thanks for your helps already...

be welll...
 

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  • #2
The Fourier transform in polar coordinates is quite similar to the Fourier transform in Cartesian coordinates. The main difference is that the integral is converted from a Cartesian double integral to a polar single integral, and the double integral is replaced with a product of two integrals.

The first integral is for the radial coordinate $\rho$ and is usually evaluated first. This can be done by substituting the Fourier transform of the given function in the integral, and then evaluating it. In this case, the Fourier transform of $f(x,y)$ is given by:
$$F(k_x,k_y) = \frac{2}{\lambda} \exp \left(-j \frac{4 \pi}{\lambda} \right) G(\theta).$$
Substituting this expression in the integral, we get:
$$\mathcal{F}[f(x,y)] = \int_0^{2 \pi} \int_0^{\infty} \frac{2}{\lambda} \exp \left(-j \frac{4 \pi}{\lambda} \rho \right) G(\theta) \rho d\rho d\theta.$$
The second integral is for the angular coordinate $\theta$ and is usually evaluated second. This can be done by substituting the
 

1) What is a Fourier integral in polar coordinates?

A Fourier integral in polar coordinates is a mathematical technique used to decompose a function into its frequency components. It involves representing a function in terms of polar coordinates, where the distance from the origin and the angle of rotation are used as variables instead of just the x and y coordinates. This allows for a more efficient representation of periodic functions.

2) How is a Fourier integral in polar coordinates different from a regular Fourier integral?

A regular Fourier integral uses rectangular coordinates, while a Fourier integral in polar coordinates uses polar coordinates. This means that the variables used to represent the function are different, which can lead to a different representation of the function and a different set of frequency components.

3) What is the purpose of using polar coordinates in a Fourier integral?

The purpose of using polar coordinates in a Fourier integral is to take advantage of the symmetry and periodicity of certain functions. By representing a function in terms of polar coordinates, it becomes easier to identify and isolate its frequency components, making it a more efficient method for analyzing periodic functions.

4) How is a Fourier integral in polar coordinates calculated?

To calculate a Fourier integral in polar coordinates, the function is first represented in terms of polar coordinates and then integrated over a specific range of angles and radii. This results in a series of coefficients, each representing a frequency component of the function. These coefficients can then be used to reconstruct the original function.

5) What are some real-world applications of Fourier integrals in polar coordinates?

Fourier integrals in polar coordinates have many practical applications, such as in signal processing, image analysis, and quantum mechanics. They are also commonly used in fields such as engineering, physics, and mathematics for data analysis and modeling. Some specific examples include sound and image compression, radio astronomy, and solving partial differential equations.

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