In the retrospective analysis of mathematics or physics, one often discovers strange phenomena or potentially deeper and more interesting results. For instance, the Fourier transform discussed in this article can be derived from a "whimsical" line of thought.
Laurent Series
We know that a function that is well-behaved at the origin can be expanded into a Taylor series: f(x)=\sum_{n=0}^{\infty}a_n x^n We notice that the powers above are all positive. Why can’t we include negative powers of x? For example, expanding \frac{\sin z}{z^2} as: \frac{1}{z}-\frac{z}{6}+\frac{z^3}{120}\dots is clearly a reasonable thing to do. Thus, by incorporating complex functions, we obtain the Laurent series for analytic functions: f(z)=\sum_{n=-\infty}^{+\infty}a_n z^n This is a bilateral expansion of the function, where: a_n=\frac{1}{2\pi i}\int_{\gamma} \frac{f(z)}{z^{n+1}}dz Here, \gamma is the integration contour |z|=\rho, \rho>0. This formula is based on the following obvious fact (where \alpha \in \mathbb{Z}): \int_{\gamma} z^{\alpha}dz=\left\{ {\begin{array}{ll} {2\pi i,} & {\alpha=-1;} \\ {0,} & {\alpha \neq -1.} \end{array}} \right.
Semi-integer Power Series
However, another question arises: why can’t we include fractional powers? How would a function like \sqrt{z} be expanded? This suggests that the Laurent series can be further extended. To illustrate this, let us include "semi-integer powers" and consider: f(z)=\sum_{n=-\infty}^{+\infty}a_n z^{n/2}
How do we find the coefficients for each term? We can transform it into a familiar Laurent series. Let z^{1/2}=\xi, then: f(\xi^2)=\sum_{n=-\infty}^{+\infty}a_n \xi^n We already know how to find the coefficients for a Laurent series, which gives: \begin{aligned} a_n=\frac{1}{2\pi i}\int_{\gamma}\frac{f(\xi^2)}{\xi^{n+1}}d\xi &=\frac{1}{2\pi i}\int_{2\gamma}\frac{f(z)}{z^{(n+1)/2}}dz^{1/2}\\ &=\frac{1}{4\pi i}\int_{2\gamma}\frac{f(z)}{z^{n/2+1}}dz \end{aligned} In this context, \gamma is a circle centered at the origin (traversed once counter-clockwise), and n\gamma denotes traversing the origin n times counter-clockwise. This new series is one of the generalizations of the Laurent series.
Once and for All
Yet, more questions arise: why can’t there be 1/3 integer powers? Why can’t there be irrational powers? Such "whimsical" ideas could be endless. Therefore, to avoid further questioning, we might as well consider all real powers. This is what "once and for all" means in this section. Consider the function: f(z)=\int_{-\infty}^{+\infty}a(x)z^x dx This encompasses the two forms mentioned above. (Discrete a_n corresponds to a(x) containing Dirac delta functions \delta(x)). Let us derive the expression for a(x). We discretize the summation, starting from the origin and summing with a step size of \Delta x: f(z)=\lim_{\Delta x \to 0} \sum_{n=-\infty}^{+\infty}a(n\Delta x)\Delta x z^{n\Delta x} Let z^{\Delta x}=\xi, substituting this in: f(\xi^{1/\Delta x})=\sum_{n=-\infty}^{+\infty}a(n\Delta x)\Delta x \xi^{n} Using the same method as the Laurent series to find the coefficients, we get: \begin{aligned}a(n\Delta x)\Delta x=\frac{1}{2\pi i}\int_{\gamma}\frac{f(\xi^{1/\Delta x})}{\xi^{n+1}}d\xi &=\frac{1}{2\pi i}\int_{\left(\frac{1}{\Delta x}\right)\gamma}\frac{f(z)}{z^{(n+1)\Delta x}}dz^{\Delta x}\\ &=\frac{1}{2\pi i}\int_{\left(\frac{1}{\Delta x}\right)\gamma}\frac{\Delta x f(z)}{z^{n\Delta x+1}}dz \end{aligned} Which is: \begin{aligned}a(x)=\frac{1}{2\pi i}\int_{\left(\frac{1}{\Delta x}\right)\gamma}\frac{f(z)}{z^{x+1}}dz \end{aligned} The integration path is a circle winding around the origin infinitely many times. Since the transformation is reversible, this becomes a mutual transformation between two functions.
Fourier Transform
Now let us switch to familiar notation. Let z=e^{-i\omega}, replace a(x) with the symbol f(x), and replace the corresponding f(z) with F(\omega). We have: F(\omega)=\int_{-\infty}^{+\infty}f(x)e^{-i\omega x} dx And: f(x)=-\frac{1}{2\pi i}\int_{-\infty}^{+\infty} \frac{F(\omega)}{e^{-i\omega(x+1)}}de^{-i\omega}=\frac{1}{2\pi }\int_{-\infty}^{+\infty} F(\omega) e^{i\omega x}d\omega The integration curve \gamma we chose is counter-clockwise, but e^{-i\omega x} is clockwise, so a negative sign must be added. This yields the Fourier transform and its inverse.
A Brief Summary
The reader’s biggest question might be why the integration limits are from negative infinity to positive infinity, rather than from 0 to positive infinity? This is indeed not easy to explain clearly; a rigorous derivation of the inverse Fourier transform still requires the steps found in "Methods of Mathematical Physics." However, tentatively speaking, from the perspective of this article, one can still provide an explanation (an explanation, not a proof) for the integration limits.
If the upper limit of integration is chosen appropriately, then the lower limit can be chosen arbitrarily. For example, if the lower limit is 0, then the upper limit must be 2\pi N, N \in \mathbb{Z} and N \to \infty. However, as it approaches infinity, adding a restriction (N \in \mathbb{Z} rather than N \in \mathbb{R}) is always uncomfortable. Since positive infinity possesses uncertainty, we might as well introduce uncertainty at the lower limit (negative infinity). In this way, because there is no fixed starting point, there is no need to impose a restriction on the endpoint. Thus, the upper and lower limits of integration can be written as positive and negative infinity. As for the flaws in the parts approaching infinity, they are naturally smoothed out during integration. (PS: This passage is quite vague, but to avoid getting bogged down in tedious technical details, I can only say this much. In fact, based on the rigorous theory of improper integrals in mathematical analysis, a proof can be given for the above statement. This mainly involves integration techniques such as \int_0^{\infty} f(x)\cos(\omega x)dx. Friends who wish to delve deeper might want to try it?)
In fact, the derivations above are largely non-rigorous. The purpose of this article is to provide a "relatively natural" understanding of the Fourier transform through this line of thought and to offer a way to connect two seemingly unrelated things (the Fourier transform and the Laurent series). Through complex functions, many branches of mathematics can be ingeniously linked. This seems to be a principle of the Designer: harmony and unity. If this is truly the case, we have a deeper reason to explore mathematics and science—that is, for the sake of beauty!
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