Inverse Fourier Transform: Impact on Integration Limits

In summary, the conversation discusses an equation that involves integration and the potential change in integration limits after an inverse Fourier transform. The attached file is not accessible, but it is suggested that the answer would likely be yes. The conversation also mentions the function and Fourier-transform equations and how the boundaries may not remain the same after the transformation. However, the transformed function should still be equivalent to the original.
  • #1
hula
3
0
an equation involves an integration. After an inverse Fourier transform of the equation, will the integration limits change? (maybe you can take a look at the attached file)

Thanks!
 

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  • #2
I can't read your attachment, but I think in general the answer would be yes.
Consider the function
[tex]F(a) = \int_0^a f(x) \, dx[/tex]
The Fourier-transform will be
[tex]\tilde F(k) = \frac{1}{\sqrt{2\pi}} \int e^{-ika} F(a) \, da
= \frac{1}{\sqrt{2\pi}} \int \int_0^a f(x) e^{-ika} \, dx \, da
[/tex]
If you are lucky you will be able to do the integral in a and are left with one other integral, whose boundaries are probably not the same. But of course, if you transform this back, you should get the original function again (otherwise it's not really a good Fourier transform, is it)
 
  • #3


The inverse Fourier transform is a mathematical operation that takes a function in the frequency domain and converts it back to the time domain. When performing an inverse Fourier transform, the integration limits do not change. This is because the integration limits are determined by the properties of the original function and not by the Fourier transform itself.

In the attached file, it can be seen that the inverse Fourier transform is represented by the integral from negative infinity to positive infinity. This is the standard form for the inverse Fourier transform and the integration limits remain the same regardless of the function being transformed.

However, it is important to note that the function being transformed may have different properties in the time and frequency domains, which can affect the integration limits. For example, a function with finite support in the time domain will have infinite support in the frequency domain and vice versa. In such cases, the integration limits may need to be adjusted to accurately represent the function in the transformed domain.

In conclusion, the inverse Fourier transform does not have an impact on the integration limits as they are determined by the properties of the original function. However, the properties of the function in the time and frequency domains may require adjustments to the integration limits for an accurate representation in the transformed domain.
 

1. What is the inverse Fourier transform?

The inverse Fourier transform is a mathematical operation that takes a function in the frequency domain and converts it back into the time domain. This allows us to analyze the different frequencies present in a signal and understand how they contribute to the overall function.

2. How does the inverse Fourier transform impact integration limits?

The inverse Fourier transform can impact integration limits in two ways. First, it can change the limits of integration in the time domain when converting a function from the frequency domain. Second, it can also change the limits of integration in the frequency domain when converting a function from the time domain.

3. Can the inverse Fourier transform be used for any function?

Yes, the inverse Fourier transform can be used for any function as long as it satisfies certain mathematical conditions, such as being continuous and having finite energy. This allows us to apply the inverse Fourier transform in a wide range of fields, including signal processing, physics, and engineering.

4. What are the practical applications of the inverse Fourier transform?

The inverse Fourier transform has many practical applications, such as signal processing, image reconstruction, and solving differential equations. It is also used in fields such as optics, acoustics, and quantum mechanics to understand the behavior of physical systems in different domains.

5. What is the relationship between the inverse Fourier transform and the Fourier series?

The inverse Fourier transform is closely related to the Fourier series, which is a way of representing a periodic function as a sum of sinusoidal functions. The inverse Fourier transform can be seen as a generalization of the Fourier series, as it allows us to analyze non-periodic functions and functions that exist over a finite interval.

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