Exponential Fourier transform

In summary, Pero K found that the exponential Fourier transform is:$$g(a)=\frac{2}{2\pi} \int_{0}^{\infty} e^{-x} e^{-iax} dx$$.
  • #1
240
23
Homework Statement
Find the exponential Fourier transform of
##f(x)=e^{-|x|}## and write the inverse transform. You should find:
$$\int_{0}^{\infty} \frac{\cos{ax}}{a^2+1} da = \frac {\pi}{2} e^{-|x|}$$
Relevant Equations
Fourier transform:
$$g(a)=\frac{1}{2\pi} \int_{-\infty}^{\infty} f(x) e^{-iax} dx$$
Inverse Transform:
$$f(x)=\int_{-\infty}^{\infty} g(a) e^{iax} da$$
From the sketch, I know that this function is an even function. So, I simplify the Fourier transform in the limit of the integration (but still in exponential form). Then, I try to find the exponential FOurier transform. Here what I get:
$$g(a)=\frac{2}{2\pi} \int_{0}^{\infty} e^{-x} e^{-iax} dx$$,
$$g(a)=\frac{1}{\pi} \int_{0}^{\infty} e^{(-x)(1+a)} dx$$,
$$g(a)=\frac{1}{\pi} \left[\frac{e^{-ix(1+a)}}{-i(1+a)} \right]^{\infty}_{0}$$.
As x approaching infinite ##e^{-ix(1+a)}## approaching zero. So,
$$g(a)=\frac{1-ia}{\pi(1+a^2)}$$.

Knowing this transform, I did the inverse transformation.
$$f(x)=\int_{-\infty}^{\infty} \frac{1-ia}{\pi(1+a^2)} e^{iax} da$$, where ##e^{iax}=\cos {(ax)} + i \sin {(ax)}##
So,
$$f(x)=\int_{-\infty}^{\infty} \frac{(1-ia)\left(\cos{ax} + i \sin {ax}\right)}{\pi(1+a^2)} da$$.

I observe that ##\frac{\sin{ax}}{1+a^2}##; ##\frac{(-a)\cos{ax}}{1+a^2}## are odd functions. But, ##\frac{\cos{ax}}{1+a^2}##; ##\frac{(a)\sin{ax}}{1+a^2}## are even functions. So,
$$f(x)=\frac{2}{\pi}\int_{0}^{\infty} \frac{\cos {ax} + a \sin {ax}}{(1+a^2)} da$$.

The sin term of the answer shouldn't be there. I have double-checked my work and still haven't find the mistake. Could you please explain how I get the answer term, in the problem statement? Thanks.
 
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  • #2
agnimusayoti said:
Then, I try to find the exponential FOurier transform. Here what I get:
$$g(a)=\frac{2}{2\pi} \int_{0}^{\infty} e^{-x} e^{-iax} dx$$,

How did you get that?
 
  • #3
I change ##e^{-|x|}## from ##-\infty## to ##\infty## becomes ##e^{-x}## from 0 to ##\infty##.
 
  • #4
agnimusayoti said:
I change ##e^{-|x|}## from ##-\infty## to ##\infty## becomes ##e^{-x}## from 0 to ##\infty##.
What about the ##e^{-iax}##?
 
  • #5
PeroK said:
What about the ##e^{-iax}##?
It should be the same, isn't it? Because of the general formula?
 
  • #6
agnimusayoti said:
It should be the same, isn't it? Because of the general formula?
It's not an even function:
$$e^{-iax} = \cos(ax) - i\sin(ax)$$
And ##\sin(ax)## is an odd function.
 
  • #7
Oh Gee. I forgot that ##f(x)## should be multiplied by ##e^{-iax}##. I will try to fix this.
 
  • Like
Likes PeroK
  • #8
I mean, an odd or even function is multiplied function; not f(x) itself.
 
  • #9
Yeah, finally I can show the solution. Thanks, Pero K for the correction!
 

1. What is an Exponential Fourier transform?

An Exponential Fourier transform is a mathematical tool used to decompose a periodic function into a sum of complex exponential functions. It is also known as the complex Fourier transform or the Fourier series.

2. How is the Exponential Fourier transform different from the regular Fourier transform?

The Exponential Fourier transform is a specialized case of the regular Fourier transform, where the function being transformed is periodic. The regular Fourier transform can be used for both periodic and non-periodic functions.

3. What is the importance of the Exponential Fourier transform in science?

The Exponential Fourier transform is important in many fields of science, including engineering, physics, and signal processing. It allows for the analysis and synthesis of periodic signals, making it useful in understanding and manipulating various systems and phenomena.

4. How is the Exponential Fourier transform calculated?

The Exponential Fourier transform is calculated using a series of complex integrals. The specific equations used depend on the periodic function being transformed, as well as the desired frequency range and resolution.

5. What are some real-world applications of the Exponential Fourier transform?

The Exponential Fourier transform has many practical applications, such as in signal processing for audio and image compression, in electrical engineering for analyzing and designing electronic circuits, and in physics for studying the behavior of waves and oscillatory systems.

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