I have been particularly interested in complex numbers lately—or more strictly speaking, the content of complex functions. One reason is that through them, we can connect seemingly unrelated concepts, embodying the concise and unified beauty of mathematics. One topic I am quite interested in is the relationship between Taylor series and Fourier series in real analysis. Both are series expansions of a function. Taylor series are used for general function expansions, where the coefficients are obtained through n-th order derivatives. Fourier series are intended for periodic functions, and their coefficients are obtained through definite integrals. In the world of real numbers, the two are unrelated. However, complex analysis tells us: they are the same thing! They are simply "projections" of the same object into the real world from different angles, creating different "images" that lead us to believe they are different things.
Let’s look directly at a magic-like operation: We know that in the world of real numbers, we have: \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots, where |x| < 1.
Now, let us extend this directly into the world of complex numbers without proof, replacing the real number x with a complex number z: \ln(1+z) = z - \frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} + \dots
The proof of this extension belongs to the details. When learning a subject, we first need to grasp the main trunk and the whole, and then deal with the branches and leaves. If we try to handle every problem perfectly from the start, it will seriously hinder our pace of thinking and may also cause us to lose our appreciation for the beauty of mathematics. Many times, analogies made out of the pursuit of "symmetry" and "unity" often lead to many correct results and are pleasing to the eye.
If we directly analogize the rules of real number operations, we have ix = \ln(e^{ix}).
Then we have: ix = \ln(e^{ix}) = \ln\left(\frac{e^{ix}+1}{e^{-ix}+1}\right) = \ln(e^{ix}+1) - \ln(e^{-ix}+1)
Substituting this into the expansion of \ln(1+z) above, we get: ix = (e^{ix} - e^{-ix}) - \frac{e^{2ix} - e^{-2ix}}{2} + \frac{e^{3ix} - e^{-3ix}}{3} - \frac{e^{4ix} - e^{-4ix}}{4} + \dots
From the formula e^{in\theta} = \cos n\theta + i \sin n\theta, we can derive \sin n\theta = \frac{1}{2i}(e^{in\theta} - e^{-in\theta}), which implies e^{in\theta} - e^{-in\theta} = 2i \sin n\theta. Thus, the formula above becomes: ix = 2i\left(\sin x - \frac{\sin 2x}{2} + \frac{\sin 3x}{3} - \frac{\sin 4x}{4} + \dots\right)
That is: \frac{1}{2}x = \sin x - \frac{\sin 2x}{2} + \frac{\sin 3x}{3} - \frac{\sin 4x}{4} + \dots
It is important to note that we started from a Taylor series and, using complex numbers as a link, derived the Fourier series of a function. This is surprisingly delightful because, in the real world, the coefficients of these two series are derived via differentiation and integration, respectively—two operations that seem completely different. If they are truly the same thing in the complex world, then we reach the following conclusion:
Complex numbers reveal a hidden connection between differentiation and integration!
In fact, this is precisely the core content of complex analysis! In future articles, Taylor series and Fourier series will be topics we discuss frequently.
The logic above also provides an idea: to obtain the Fourier series expansion of a function f(x), we can replace x with x = -i[\ln(e^{ix}+1) - \ln(e^{-ix}+1)]. This transforms the function into the form f(-i[\ln(p+1) - \ln(q+1)]), which can then be expanded as a bivariate Taylor series in terms of p and q. Since Fourier coefficients are typically calculated using definite integrals, this seemingly provides a simple method for calculating definite integrals?? After all, calculating derivatives is much simpler than calculating integrals... Seeing this example, many fascinating ideas emerge, making us very excited. However, the reality is not yet that ideal. Our goal is to expand it into a combination of powers of e^{ix} and e^{-ix}, but for just the e^{ix} term alone, there are infinitely many terms added together (any p^{n+1}q^n term is an e^{ix} term). This is somewhat complicated to explain; interested readers can try it themselves to understand why.
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