Solving FT of Decaying Exp: Lorentzian Function

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In summary, the author showed that when t goes to infinity, the integral of a Lorentzian function equals the sum of the integrals of its terms, with a positive sign.
  • #1
Amok
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I had to show that:

[tex] \int_{-\infty}^{\infty} e^{-a \left | t \right |} e^{\omega t} \mathrm{d} t [/tex]

with a positive, is equal to a Lorentzian function.

I simply did this:

[tex]

\int_{-\infty}^{\infty} e^{-a \left | t \right |} e^{i\omega t} \mathrm{d} t = \int_{0}^{\infty} e^{(-a + i\omega) t} \mathrm{d} t + \int_{-\infty}^{0} e^{(a + i\omega)t } \mathrm{d} t
= \frac{-1}{i\omega\ -a} + \frac{1}{i\omega + a} + \lim_{x \to -\infty} e^{(-a + i\omega )x} - \lim_{y \to +\infty} = \frac{2a}{\omega^2 + a^2} + \lim_{x \to +\infty} e^{-ax} lim_{x \to +\infty} e^{i\omega} - \lim_{y \to -\infty} e^{ax} \lim_{y \to -\infty} e^{i\omega} = \frac{2a}{\omega^2 + a^2} [/tex]

Is this correct? I'm asking because I've seen people do this by using Euler's formula and then calculating an integral with a cosine in it, but I don't really see the point of that.

EDIT: can you guys see the whole thing? I can't, what should I do?
 
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  • #2
You're fine up to here:

[tex]\frac{-1}{i\omega\ -a} + \frac{1}{i\omega + a} + \lim_{t \to \infty} e^{(-a+i\omega)t} - \lim_{t \to -\infty} e^{(a+i\omega)t}[/tex]

Then you just need to say the two limits are equal to 0.

What you can't do is split the limits up like you did:

[tex]\lim_{t \to \infty} e^{(-a+i\omega)t} \Rightarrow \lim_{t \to \infty} e^{-at}\lim_{t \to \infty} e^{i\omega t}[/tex]

You can only say lim AB = (lim A)(lim B) if both lim A and lim B exist. In this case, the complex exponential oscillates and doesn't converge as t goes to infinity, so the limit doesn't exist.
 
  • #3
Or, to add a little detail:

|e(-a+iω)t| = |e-at||eiωt| = |e-at| → 0 as t → ∞

and similarly for the other one.
 
  • #4
vela said:
[tex]\lim_{t \to \infty} e^{(-a+i\omega)t} \Rightarrow \lim_{t \to \infty} e^{-at}\lim_{t \to \infty} e^{i\omega t}[/tex]

You can only say lim AB = (lim A)(lim B) if both lim A and lim B exist. In this case, the complex exponential oscillates and doesn't converge as t goes to infinity, so the limit doesn't exist.

That's what I thought, so how do I show that the limit does exist?
 
  • #5
Do what LCKurtz did.
 
  • #6
But how did he introduce the modulus in the calculations?
 
  • #7
If |z| → 0 then doesn't that show z → 0?
 
  • #8
Of course it does! Thank you guys.
 
  • #9
LCKurtz said:
If |z| → 0 then doesn't that show z → 0?

I'm sorry to bring thus up again, but I can see that what you're saying is true, simply by picturing the polar form of a complex number on a complex plane, but how do you prove what you're saying? How can you really write it down?
 
  • #10
Doesn't z → 0 mean |z - 0| gets small?
 
  • #11
It mosta certainly does. The modulus of my function is

[tex] \left| e^{-at} \right| [/tex]

which does become arbitrarily small as t goes to infinity. I hadn't given much though to limits of complex functions for a while. This was good practice. Thank you very much Kurtz.
 

1. What is the Lorentzian function in the context of solving Fourier transforms of decaying exponential functions?

The Lorentzian function is a mathematical function used to describe the shape of spectral lines in spectroscopy and the resonance curve in physics. In the context of solving Fourier transforms of decaying exponential functions, it is used to model the decay rate of a signal over time.

2. How is the Lorentzian function related to the Fourier transform of a decaying exponential function?

The Fourier transform of a decaying exponential function is a complex-valued function that represents the frequency components of the original signal. The Lorentzian function is used to describe the decay rate of these frequency components, allowing us to better understand the behavior of the signal over time.

3. What is the significance of solving the Fourier transform of a decaying exponential function?

Solving the Fourier transform of a decaying exponential function allows us to extract useful information about a signal, such as its frequency components and decay rate. This can be applied in various fields, such as signal processing, physics, and engineering.

4. What are some common techniques for solving the Fourier transform of a decaying exponential function?

Some common techniques for solving the Fourier transform of a decaying exponential function include the use of complex analysis, Laplace transforms, and the convolution theorem. These techniques allow us to convert the time-domain signal into the frequency domain, where we can more easily analyze its behavior.

5. Can the Lorentzian function be used to model other types of decaying signals?

Yes, the Lorentzian function can be used to model other types of decaying signals besides decaying exponential functions. It is commonly used to describe the behavior of decaying oscillating signals, such as those found in damped harmonic oscillators.

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