The domain of the Fourier transform

In summary, the conversation discusses the possibility of extending the Fourier transform to complex valued functions of a complex variable by using a double integral over the real line. This could have potential theoretical or practical applications, and it is similar to the two-dimensional Fourier transform used for images. However, it is important to note that transforming a purely imaginary variable is equivalent to transforming the real version of the variable rotated π/2, which leads to the Laplace transformation.
  • #1
redtree
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TL;DR Summary
How does the Fourier transform handle functions of complex variables when only integrating over the reals?
Given the domain of the integral for the Fourier transform is over the real numbers, how does the Fourier transform transform functions whose independent variable is complex?

For example, given
\begin{equation}
\begin{split}
\hat{f}(k_{\mathbb{C}}) &= \int_{\mathbb{R}} f(z_{\mathbb{C}}) e^{2 \pi i k_{\mathbb{C}} z_{\mathbb{C}}} d z_{\mathbb{C}}
\end{split}
\end{equation}
where ##z_{\mathbb{C}} = z_1 + z_2 i## and ##k_{\mathbb{C}} = k_1 + k_2 i##.

It seems that evaluating ##f(z_{\mathbb{C}}) e^{2 \pi i k_{\mathbb{C}} z_{\mathbb{C}}}## over the reals includes only ##z_1## and ignores ##z_2 i## in ##f(z_{\mathbb{C}}) ##.
 
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  • #2
The way I 've been taught Fourier transform is that it applies to complex valued functions of a real variable and it gives as result also a complex valued function of a real variable.

That is, it is ##f:\mathbb{R}\to\mathbb{C}## and its Fourier transform ##\hat f:\mathbb{R}\to\mathbb{C}## so the integrals of the Fourier and the inverse Fourier transform are inevitably over the real line.

The integral you are proposing here seems to be an interesting generalization of the Fourier transform so that it can include functions ##f:\mathbb{C}\to\mathbb{C}##, that is complex valued functions of a complex variable. Only thing is that you have to make it a contour integral in order to make it integrate over the complex plane. Or simply a double integral over the real line like for example
$$\int_\mathbb{R}\int_\mathbb{R}f(z_1+z_2i)e^{2\pi i k_{\mathbb{C}}(z_1+z_2i)}dz_1dz_2$$

I can't tell if such an integral has interesting theoretical or practical applications.
 
  • #3
There is such a thing as a two-dimensional Fourier transform, for instance the spatial Fourier transform of an image. It consists of Fourier transforms applied independently in the x and y directions.

Perhaps that's what you need to do here, transform Re(z) and Im(z) to Re(k) and Im(k).
 
  • #4
... and remember that [itex]i=e^{\frac{i\pi}{2}} [/itex], which means that the transformation of a purely imaginary variable is the same as the transformation of the real version of the variable rotated π/2. This leads to the Laplace transformation.
 

Related to The domain of the Fourier transform

1. What is the Fourier transform?

The Fourier transform is a mathematical operation that decomposes a function or signal into its constituent frequencies. It is used to analyze and understand the frequency components of a signal, which can be useful in a variety of scientific and engineering fields.

2. How is the Fourier transform used in science?

The Fourier transform is used in a wide range of scientific disciplines, including physics, engineering, and mathematics. It is used to analyze signals in fields such as signal processing, image processing, and quantum mechanics. It is also used in solving differential equations and in the study of wave phenomena.

3. What is the difference between the Fourier transform and the inverse Fourier transform?

The Fourier transform converts a function from the time or spatial domain to the frequency domain, while the inverse Fourier transform converts it back from the frequency domain to the time or spatial domain. They are essentially inverse operations of each other and are used together to analyze and manipulate signals.

4. Can the Fourier transform be applied to any function?

The Fourier transform can be applied to any function that is integrable, meaning that it has a finite integral over its domain. However, the function must also satisfy certain conditions, such as being continuous and having a bounded number of discontinuities, in order for the transform to be well-defined.

5. Are there any limitations to the Fourier transform?

The Fourier transform has some limitations, such as the uncertainty principle, which states that it is impossible to simultaneously know the exact frequency and time information of a signal. It also assumes that the signal is periodic, which may not always be the case in real-world applications. Additionally, the Fourier transform can only be applied to functions that are integrable, as mentioned in the previous question.

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