In the very first line the limits of the first integration are those for the Fourier sine transformation ( 0 to infinity), but the integrating factor is the exponential function.
The Fourier exponential transformation should be negative infinity to positive infinity.
I have to tell you that the standard of your mathematics is a long way from up to understanding or using Fourier transforms. This is Final undergraduate and into graduate level stuff at university.
To expand on what PMB says,
Fourier transforms are one of many types of integral transforms that have very wide application in applied mathematics, optics, heat, hydrodynamics, elasticity,electric circuit theory, electromagnetism, signal and communication theory and geophysics to name but a few.
Integral transforms always come in pairs the transform and the inverse transform. They are useless without both parts.The most common in electrical theory is the Laplace transform.
The idea of a transform is to make the solution of an equation easier to solve by transforming it into another variable.
You then solve the easier equation in the other variable.
But then the price for this you pay is you have to transform the solution back to the original variable, hence the need for the inverse transform.
You may have done logs in school, this is a transform method for multiplying to numbers, a and b so that
ab may be difficult but log(ab) = log(a) + log(b)
So the log transforms a multiplication into an addition, but to get the actual product you need to take the antilog.
Last comment, there is something called the Fast Fourier Transform (FFT). This is something quite different and should not be confused with the Fourier transform. The FFT is basically a numerical method, suitable for computers, of spectral or harmonic analysis of data points.
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Last edited by studiot; Jul 23rd 2017 at 03:37 AM.
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