What does this newly arranged equation for Fourier Transform mean?

Here's an equation of the Fourier transform: Fourier Equation

In the same equation image above, in the second line, is an equation I read from a paper. It seems that the a_n and b_n are turned into functions of A(u) and B(u) respectively, which depend on the variable u, which is the frequency. There is also an additional 2*pi in the sin and cos functions.

In other words, the eventual equation can get to become: New Equation

I don't understand several things in this new equation.
First, what are the A(u) and B(u) functions? Why do they depend on u?
Second, why is there a 2*pi in the sin and cos functions? What do the 2*pi mean?
Third, how does the new equation differ from the original equation?
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Commented:
a correct formulation (which might have nothing to do with your problem) would be

\frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos(nx) + b_n \sin(nx)
above line not rendered very well. Try
f(x) = a_0/2 + sum (from n = 0 to infinity)(a_n cos(nx) + b_n sin(nx)
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Commented:
A(u) and B(u) are just functions rather than constants. It depends on the problem
The 2 pi  is just 2 x 3.14159... and are used to change the frequency variable from radians to cycles per second.
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Author Commented:
@aburr, Thanks. Just to clarify a little, since u is the frequency, is there any special reason/occasions when one would choose to have the functions A(u) and B(u), which are replacing the constants, to depend on u?
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Commented:
Where did the summation index  n go?
Could they have meant A(n) where they wrote A(u)?
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Commented:
I thnk ozo has hit upon a fundamental difficulty
The second line is  missing more than just the missing n. The last line it should have a cos or a phase angle.
Also if u is a frequency, what is x?  It cannot be a distance.
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Author Commented:
Sorry about the variable n in the summation. It should be u. I had a typo.

I believe x refers to the time in the time domain.
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Commented:
"Sorry about the variable n in the summation. It should be u. I had a typo. "
I am still confused.
which n's should be u's? Or which u's should be n's.    If  you sum over u it cannot be a frequency. n is not a variable, it is an index.
Saying x is a time clears up a number of points
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Commented:
Can you restate the equations with the typos corrected?
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Author Commented:
There shouldn't be n in the equation. It was a typo.

This is the correct equation: Corrected Equation

The variable x is referring to the time domain, and the variable u is referring to the frequency domain.

Sorry about the confusion. Hope this clarifies and helps to answer my question better.
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Commented:
There is still something seriously wrong with the equation.
If u is a frequency, what is the difference in cycles per second between u = 5 and u = 7?
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The indices of a summation are usually integers and dimensionless, in which case the dimensionality of the arguments of sin are incorrect
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Author Commented:
I'm not quite sure what is wrong. In this case, how should a correct Fourier transform probably look like if it has to be in this "sort of arrangement"?
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Commented:
The last three lines of my post told exactly some of the things wrong
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Author Commented:
Thanks. The latex doesn't render on EE, but works fine when I paste it on a latex editor. The latex form actual reads better. Thanks! :)
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