The Answer Solving Fourier Transforms: Odd & Even Functions

• Niles
In summary, the conversation discusses solving a problem related to Fourier transforms. The person asking the question has shown that the Fourier transform of a specific function is equal to a certain equation. They then bring up a question about the inverse Fourier transform and how to rewrite it when an odd function is multiplied by an even function. The person eventually finds the solution by reading about the inverse Fourier transform of an even function.
Niles
[SOLVED] Fourier transforms

Homework Statement

Please take a look at the following:

I have shown that the Fourier transform of f(t) = exp(-|t|) = $$\sqrt{\frac{2}{pi}}\cdot \frac{1}{1+\omega ^2}$$.

Now I am having trouble with question A. I know what the inverse Fourier transform is given by, but the we have an odd function (exp(iwt)) multiplied to an even function (the above). This results in an odd function, so how do I rewrite it?

Ok, I read something about the inverse Fourier transform of an even function, and it adds up now. Problem solved.

1. What is a Fourier transform?

A Fourier transform is a mathematical tool used to decompose a function into its individual frequency components. It converts a function from its original domain (usually time or space) to a representation in the frequency domain, allowing for analysis and manipulation of the function's frequency components.

2. What is the difference between odd and even functions?

An odd function is symmetric about the origin, meaning that if you reflect the function across the y-axis, it will look the same as the original. In contrast, an even function is symmetric about the y-axis, meaning that if you reflect the function across the y-axis, it will look the same as the original. This symmetry can be seen in the Fourier transform as well.

3. How do you solve Fourier transforms for odd and even functions?

To solve a Fourier transform for an odd or even function, you can use the properties of odd and even functions to simplify the integral. For an odd function, the integral will be zero if the limits of integration are symmetric about the origin. For an even function, the integral will be twice the integral from 0 to infinity. This allows for an easier integration process and ultimately, a simpler Fourier transform.

4. Why is it important to identify whether a function is odd or even before solving its Fourier transform?

Identifying whether a function is odd or even is important because it can greatly simplify the process of solving its Fourier transform. As mentioned before, this identification allows for the use of special properties that make the integration process easier and the final transform simpler.

5. Can Fourier transforms be used in other fields besides mathematics?

Yes, Fourier transforms have a wide range of applications in various fields such as physics, engineering, and signal processing. They are commonly used in image and sound processing, as well as in the analysis of vibrations and waves. They also play a crucial role in quantum mechanics and other areas of physics.

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