Using Fourier Transforms to solve Definite Integrals with Limits 0 to Infinity

In summary, the conversation discusses using Fourier Transforms to solve Definite Integrals with limits from 0 to Infinity. The person is struggling to understand how to solve these types of integrals and is seeking help. The conversation also mentions the use of even and odd functions in solving the problem.
  • #1
1. Using Fourier Transforms to solve Definite Integrals with Limits 0 to Infinity

I'm trying to understand how to use Fourier Transforms to solve Definite Integrals with limits from 0 to Infinity.
I understand how to use Fourier Transforms to solve indefinite integrals, but I believe there is supposed to be a much simpler way to solve these definite integrals without working out the entire indefinite solution.


Homework Equations


X(j0) = Integral from -Infinity to Infinity : x(t)dt
x(0) = Integral from -Infinity to Infinity: X(jw)dw
 
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  • #2
Welcome to PF! Here's a hint: Think about even and odd time functions.
 
  • #3
[tex]\displaystyle\int^\infty_0 2te^-^a^t\cos(t)\,dt = ?}[/tex]

I'm not entirely sure how to manipulate this.. the [tex]e^-^a^t[/tex] makes it so the function is neither even nor odd anymore.. even if I pulled the even part out to rewrite the function so its integrated from -infinity to infinity I'm unsure about what to do with the odd piece.
 
  • #4
Yikes! I don't know what to do with this either. Sorry I jumped in...
 
  • #5
marcusl said:
Yikes! I don't know what to do with this either. Sorry I jumped in...

I appreciate the thought! Thanks! :)
 

1. What is a Fourier Transform?

A Fourier Transform is a mathematical operation that decomposes a function into its constituent frequencies. It is commonly used in signal processing and can be used to solve certain types of integrals.

2. How does a Fourier Transform help to solve definite integrals with limits 0 to infinity?

A Fourier Transform allows us to transform a function from the time or spatial domain to the frequency domain. This transformation can simplify the integration process, making it easier to solve integrals with infinite limits.

3. Can all definite integrals with limits 0 to infinity be solved using Fourier Transforms?

No, not all integrals can be solved using Fourier Transforms. Fourier Transforms are most useful for integrals that involve oscillatory functions or those that can be represented by a sum of sinusoidal functions.

4. Are there any limitations or challenges when using Fourier Transforms to solve definite integrals with limits 0 to infinity?

One limitation is that the function must be smooth and well-behaved for the Fourier Transform to be applicable. Additionally, some integrals may require more advanced techniques in combination with Fourier Transforms.

5. Can Fourier Transforms be used in other areas of science besides solving integrals?

Yes, Fourier Transforms have many applications in various fields of science, including signal processing, image processing, and quantum mechanics. They are also used in practical applications such as audio and video compression.

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