What is the Exponential Fourier Transform of an Even Function?

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
agnimusayoti
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!
 

Related to What is the Exponential Fourier Transform of an Even Function?

1. What is an Exponential Fourier transform?

The Exponential Fourier transform is a mathematical tool that decomposes a periodic function into a sum of complex exponential functions. It is commonly used in signal processing and analysis to convert a signal from the time domain to the frequency domain.

2. How is an Exponential Fourier transform calculated?

The Exponential Fourier transform is calculated by integrating the original function multiplied by a complex exponential function over one period of the function. This process is repeated for each frequency component in the function, resulting in a series of complex numbers that represent the amplitude and phase of each frequency component.

3. What is the difference between an Exponential Fourier transform and a Fourier series?

An Exponential Fourier transform is a continuous version of a Fourier series, which is a representation of a periodic function as a sum of sinusoidal functions. The main difference is that a Fourier series only considers real-valued sinusoidal functions, while an Exponential Fourier transform allows for complex-valued exponential functions.

4. What are the applications of an Exponential Fourier transform?

An Exponential Fourier transform has many applications in engineering, physics, and mathematics. It is commonly used in signal processing, circuit analysis, and image processing to analyze and manipulate signals in the frequency domain. It is also used in solving differential equations and in quantum mechanics.

5. What are the limitations of an Exponential Fourier transform?

An Exponential Fourier transform is only applicable to periodic functions, meaning it cannot be used for non-periodic signals. It also assumes that the function is continuous and has a finite number of discontinuities. Additionally, it may not accurately represent functions with sharp changes or spikes, as it relies on a series of smooth exponential functions.

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