Fourier representation of aperiodic irregular function

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SUMMARY

The discussion focuses on the Fourier representation of an aperiodic irregular function, specifically analyzing the conditions under which the function's epsilon values are defined. The equations provided detail the behavior of the epsilon function across different ranges of z, leading to the formulation of the corresponding Fourier transform. The transform is expressed as a double integral involving exponential terms and epsilon values, which are critical for understanding the function's behavior in various domains.

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  • Familiarity with complex analysis, particularly in relation to integrals and exponential functions.
  • Knowledge of aperiodic functions and their mathematical properties.
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MadMax
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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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a small hint please?
 

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