Fourier representation of aperiodic irregular function

In summary, the conversation discusses the equations for \epsilon(i f, r) for different ranges of z, and the corresponding Fourier transform. The Fourier transform can be shown using the equation \frac{i}{q_z} \int d^2x e^{iq_\bot \cdot x}[\epsilon_2 e^{iq_z[H+h_2(x)]} - \epsilon_1 e^{iq_z h_1(x)}]. Assistance or guidance in showing this equation is requested.
  • #1
MadMax
99
0
We have

[tex]\epsilon(i f, r) = \epsilon_2(i f)[/tex] when [tex]H + h_2(x) \leq z < + \infty[/tex]
[tex]\epsilon(i f, r) = 0[/tex] when [tex]h_1(x) < z < H + h_2(x)[/tex]
[tex]\epsilon(i f, r) = \epsilon_1(i f)[/tex] when [tex]- \infty < z \leq h_1(x)[/tex]

show the corresponding Fourier transform is

[tex]\frac{i}{q_z} \int d^2x e^{iq_\bot \cdot x}[\epsilon_2 e^{iq_z[H+h_2(x)]} - \epsilon_1 e^{iq_z h_1(x)}][/tex]

I've looked in a few books but tbh I have no real idea how to show this...

Any help/suggestions/tips would be much appreciated. Thanks.
 
Last edited:
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  • #2
a small hint please?
 

FAQ: Fourier representation of aperiodic irregular function

What is Fourier representation of aperiodic irregular function?

Fourier representation of aperiodic irregular function is a mathematical technique used to decompose a function into a combination of simple sinusoidal functions. This representation is particularly useful for analyzing aperiodic and irregular functions that cannot be represented by a single equation.

How is Fourier representation of aperiodic irregular function different from Fourier series?

Fourier representation of aperiodic irregular function is different from Fourier series in that it can be used for functions that are not periodic. Fourier series can only be used for functions that repeat themselves over and over again in a regular pattern.

What is the mathematical formula for Fourier representation of aperiodic irregular function?

The mathematical formula for Fourier representation of aperiodic irregular function is given by: F(x) = a0 + ∑an cos(nx) + ∑bn sin(nx), where a0, an, and bn are coefficients determined by the function's amplitude and frequency.

Can any aperiodic irregular function be represented using Fourier representation?

Yes, any aperiodic irregular function can be represented using Fourier representation. However, the accuracy of the representation depends on the number of terms used in the formula. As the number of terms increases, the representation becomes more accurate.

How is Fourier representation of aperiodic irregular function used in real-world applications?

Fourier representation of aperiodic irregular function has many practical applications, such as signal processing, image compression, and data analysis. It is also commonly used in fields like physics, engineering, and finance to model and analyze complex systems.

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