# Messy Fourier Transform Integral

• TimH
In summary, the Fourier transform of a function is given by g(a)=-sin(a)/(pi*a), where g(a) is the Fourier representation of f(x). The Fourier representation can be found by integrating over the function from - to + infinity, using the formula g(a)*e^(iax). The second term in the integral is a function of a and x, but is dropped if it's an odd function.
TimH
I'm teaching myself some basic Fourier analysis from Boas's "Mathematical Methods in the Physical Sciences." I'm a little stuck on Example Problem 1 on p.382. This is a basic example of getting the g(a) Fourier transform of a certain function (pictured on the page) and then plugging it into the formula to get the FT of f(x). The problem has two typos in it according to the online errata, which I've taken into account.

Anyway, you get the g(a) function which is g(a)=sin(a)/(pi*a) (I'm using "a" for the alpha that appears in the book). I understand this.

Then you want to get your Fourier representation of f(x) so you plug g(a) into the integral from - to + infinity of g(a)*e^(iax) da. This I understand.

Now for this integral she expands e^(iax) using Euler to get cos(ax)+i*sin(ax). If you multiply this by g(a), i.e. by sin(a)/(pi*a), you get two terms in your integral, i.e. the integral is of (sin(a)(cos ax))/(pi*a) + (sin(a)(i*sin(ax))/(pi*a) da.

But she does not show these two terms of the multiplication. Rather, she says that the integral is equal to two times the integral of the first term, because the function "sin(a)/a is an even function." So the complex second term is dropped. This I don't understand. The second term is a function of a and x, times i. I can't really tell if its even or odd. If it's odd, I can see dropping it from the integral since the integral is symmetric around the origin so the integral of the term will be = 0. But the fact that its got an "i" in it is throwing me, too.

Could somebody please help me with this second term in the integral? Thanks.

A function of two variables has two parity properties. For example y*cos(x) is even wrt x but odd wrt y. So when you integrate over the function wrt to only one variable, it's the parity wrt to that variable which counts.
For example sin(a*x) is odd wrt a (even wrt x) so integrating over a symmetric interval wrt a will result in zero.

Okay I think I get it. The integral is of i*(sin(a)*sin(ax))/a. We're integrating wrt a. Now sin(a)/a is even. But sin(ax) is odd if we're integrating wrt a. So its an even function times an odd function which gives an odd, and so its integral is zero over the symmetric interval. And so the "i" nicely goes away.

Thanks!

## What is a Messy Fourier Transform Integral?

A Messy Fourier Transform Integral is a mathematical tool used to analyze data in the frequency domain. It is a complex integral that involves the Fourier transform, which transforms a signal from the time domain to the frequency domain.

## How is a Messy Fourier Transform Integral different from a regular Fourier Transform?

A Messy Fourier Transform Integral is more complex and involves additional mathematical operations, such as integration. It is typically used to analyze signals that are not well-behaved or have noise, while a regular Fourier Transform is used for more simple and well-behaved signals.

## What is the purpose of using a Messy Fourier Transform Integral?

The purpose of using a Messy Fourier Transform Integral is to analyze signals in the frequency domain, which can provide insight into the underlying patterns and structures of the data. It is particularly useful for analyzing signals that are not smooth or have noise, as it can help identify important features and filter out noise.

## What types of signals are suitable for a Messy Fourier Transform Integral?

A Messy Fourier Transform Integral is suitable for signals that are not well-behaved or have noise, such as signals with sudden changes or spikes, non-periodic signals, and signals with missing data points. It is also useful for analyzing signals with non-stationary behavior, meaning the signal changes over time.

## How is a Messy Fourier Transform Integral used in scientific research?

A Messy Fourier Transform Integral is commonly used in fields such as physics, engineering, and biology to analyze data and extract important information in the frequency domain. It is also used in signal processing to filter out noise and improve the quality of signals. In scientific research, it can provide valuable insights into the underlying patterns and structures of data, allowing for a deeper understanding of the phenomena being studied.

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