Editing Fourier transform (section)

=== Complex sinusoids ===
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The red [[sine wave|sinusoid]] can be described by peak amplitude (1), peak-to-peak (2), [[root mean square|RMS]] (3), and [[wavelength]] (4). The red and blue sinusoids have a phase difference of {{mvar|θ}}.
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In general, the coefficients <math>\widehat f(\xi)</math> are complex numbers, which have two equivalent forms (see [[Euler's formula]]):
<math display="block"> \widehat f(\xi) = \underbrace{A e^{i \theta}}_{\text{polar coordinate form}}
= \underbrace{A \cos(\theta) + i A \sin(\theta)}_{\text{rectangular coordinate form}}.</math>

The product with <math>e^{i 2 \pi \xi x}</math> ({{EquationNote|Eq.2}}) has these forms:
<math display="block">\begin{aligned}\widehat f(\xi)\cdot e^{i 2 \pi \xi x}
&= A e^{i \theta} \cdot e^{i 2 \pi \xi x}\\
&= \underbrace{A e^{i (2 \pi \xi x+\theta)}}_{\text{polar coordinate form}}\\
&= \underbrace{A\cos(2\pi \xi x +\theta) + i A\sin(2\pi \xi x +\theta)}_{\text{rectangular coordinate form}}.\end{aligned}</math>
which conveys both [[amplitude]] and [[phase offset|phase]] of frequency <math>\xi.</math> Likewise, the intuitive interpretation of {{EquationNote|Eq.1}} is that multiplying <math>f(x)</math> by <math>e^{-i 2\pi \xi x}</math> has the effect of subtracting <math>\xi</math> from every frequency component of function <math>f(x).</math><ref group="note">A possible source of confusion is the [[#Frequency shifting|frequency-shifting property]]; i.e. the transform of function <math>f(x)e^{-i 2\pi \xi_0 x}</math> is <math>\widehat{f}(\xi+\xi_0).</math>&nbsp; The value of this function at &nbsp;<math>\xi=0</math>&nbsp; is <math>\widehat{f}(\xi_0),</math> meaning that a frequency <math>\xi_0</math> has been shifted to zero (also see [[Negative frequency#Simplifying the Fourier transform|Negative frequency]]).</ref> Only the component that was at frequency <math>\xi</math> can produce a non-zero value of the infinite integral, because (at least formally) all the other shifted components are oscillatory and integrate to zero. (see {{slink||Example}})

It is noteworthy how easily the product was simplified using the polar form, and how easily the rectangular form was deduced by an application of Euler's formula.