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I'm not very familiar with more advanced concepts, but I have a decent understanding of single variable calculus with real-valued functions $f: \mathbb R \to \mathbb R$ (via Spivak). I wanted to learn a little bit about the Fourier Transform without going through the proper introductory material...and hit a bit of a snag. Specifically, I am trying to understand how to properly visualize the integral $\displaystyle\int_{-\infty}^{\infty}s(t)e^{-i\omega t}dt$. From a little reading, the easiest signal to work with seems to be $s(t)=e^{i\omega_0 t}$. Then, if we let $\Delta \omega=\omega_0-\omega,$ we can rewrite the integral as:

$$\int_{-\infty}^{\infty}s(t)e^{-i\omega t}dt=\int_{-\infty}^{\infty}e^{i\omega_0 t}e^{-i\omega t}dt=\int_{-\infty}^{\infty}e^{i\Delta \omega t}dt$$

For no particular reason (other than to establish the appropriate corresponding graph), suppose $\Delta \omega \gt 0$ (i.e. $\omega \neq \omega_0$). Accordingly, we can view the resulting function $g(t)=e^{i\Delta \omega t}$ (I think) as $g: \mathbb R \to \mathbb C$ with the following graph: Sinusoidal Spiral

Going forward from here I do not think I really have the terminology to properly communicate, but I think my above depiction of $g$ is using an implicit bijection between $a+bi \in \mathbb C$ and $(a,b) \in \mathbb R^2$ (hopefully, that is okay...I know the are some distinctions between $\mathbb C$ and $\mathbb R$ when it comes to specific operations).

So, assuming the graphical depiction makes enough sense, I wondered how best to think of the integral of $g$ with respect to the $T$ axis. I have several ideas (e.g. a 2D projection plane whose origin passes through the $T$ axis and rotates about the $T$ axis at the same rate as $\Delta \omega$ such that the norm of the projection plane is always 'facing' the spiral's location at time $t$) as to how I could visualize this integral, but I simply do not know.

From reading about Fourier Transforms, I know that $F(\omega)=\int_{-\infty}^{\infty}g(t)dt=0$, so I think I can definitely rule out a volume integral interpretation that is bounded by the sprial.

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    $\begingroup$ You could also use Euler's formula to write $$\int_{-\infty}^{\infty}e^{i\Delta \omega t}dt=\int_{-\infty}^\infty \cos(\Delta \omega t)~dt+i\int_{-\infty}^\infty \sin(\Delta \omega t)~dt$$ to get two real single-variable integrals. Also, the function $s(t)=e^{i\omega_0 t}$ is a bit of a problematic example to start with: the Fourier transform is $\delta(\omega-\omega_0)$, which is not a function in the usual sense. $\endgroup$
    – Semiclassical
    Commented Jun 29 at 16:57
  • $\begingroup$ @Semiclassical thank you for providing that result. I am actually a little surprised that that is how integration is defined on / extended to $\mathbb C$. Is it generally true that if $f: \mathbb R \to \mathbb C$ and $f(t) \mapsto g(t)+i h(t)$, then $\int g(t)dt=\int f(t) dt + i\int h(t)dt$? This result makes me think that integration extended to $\mathbb C$ sort of adds this notion of 'area with a specific direction' (in this case, a $\mathbb R$eal direction and a $\mathbb C$omplex direction). $\endgroup$
    – S.C.
    Commented Jun 29 at 17:18
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    $\begingroup$ If you're thinking of integration in the sense of Riemann integration then it's not too surprising: a Riemann sum of complex numbers splits into two Riemann sums for the real and imaginary parts. (For other definitions of integral, this splitting into real/imaginary parts is indeed how a complex-valued integrand is defined in the first place. See for instance definition 3.5 here for the case of Lebesgue integration.) $\endgroup$
    – Semiclassical
    Commented Jun 29 at 17:30
  • $\begingroup$ @Semiclassical Is it reasonable to conceptualize this as...$f$'s projection onto the real plane (call it $g$) would represent one leg of a 'triangle' and $f$'s projection onto the imaginary plane (call it $h$) would represent the other leg of a 'triangle'. Taking the Riemann integral of each projection then provides a magnitude + a direction (for each)...so, really, the actual integral is some notion of $\overrightarrow{\int f(t) dt}=\overrightarrow{\int g(t) dt}+\overrightarrow{\int h(t) dt}$, where $\overrightarrow{\int g(t) dt}$ is a vector with an orientation along the $\mathbb R$eal axis $\endgroup$
    – S.C.
    Commented Jun 29 at 18:57
  • $\begingroup$ and $\overrightarrow{\int h(t) dt}$ is a vector with an orientation along the $\mathbb C$omplex axis. Accordingly, (for example) for some fixed lower bound $\alpha$, $\displaystyle\int_\alpha^{\tau} f(t) dt$ is really outputting a scaled vector (belonging to $\mathbb C$) that tells you something about the accumulated 'direction' of all of the vector-like complex numbers $z$ for each argon plane you passed through (starting at $\alpha$) to arrive at $\tau$. $\endgroup$
    – S.C.
    Commented Jun 29 at 18:57

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