Complex Analysis Study 1: Revealing the Connection Between Differentiation and Integration

复分析学习1：揭示微分与积分的联系 苏剑林 · 2012-08-02 · Mathematics complex number analysis complex analysis

I've been particularly interested in complex numbers lately—or, to be more precise, in functions of a complex variable—partly because they let us tie together things that seem completely unrelated, which really shows off the simplicity and unity that mathematics can have. One thing I find especially fascinating is the pair formed by Taylor series and Fourier series in real analysis. Both are series expansions of a function: the Taylor series applies to general functions and its coefficients are obtained by taking n-th order derivatives; the Fourier series applies to periodic functions and its coefficients are obtained via definite integrals. In the real world these two seem to have nothing to do with each other, but complex analysis tells us something remarkable: they are actually the same thing! It's just that, when "projected" onto the real world from different angles, they produce different "images", which makes us think they're different objects when they're not.

Let's jump straight into a bit of seeming magic.

We know that, in the real world, we have

$ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+...$, where $|x| < 1$more

Now, without proof, let's extend this directly into the complex world, i.e., replace the real number x with a complex number z.

$$ln(1+z)=z-\frac{z^2}{2}+\frac{z^3}{3}-\frac{z^4}{4}+...$$

Working out a rigorous proof of this extension is a matter of detail, and when we're first learning a subject, what matters most is grasping the trunk and the overall shape of the tree, before worrying about the branches and leaves. If we insist on getting every single detail flawlessly right from the very start, it will seriously slow down our thinking, and may even make us lose our appreciation for the beauty of mathematics. Very often, analogies made in pursuit of "symmetry" and "unity" already lead us to many correct results, and are a genuine pleasure to behold.

If we directly carry over the rules of real-number arithmetic by analogy, we have $ix=ln(e^{i x})$,

which gives us $ix=ln(e^{i x})=ln(\frac{e^{ix}+1}{e^{-ix}+1})=ln(e^{ix}+1)-ln(e^{-ix}+1)$.

Substituting this into the expansion of $ln(1+z)$ above, we get

$$ix=(e^{ix}-e^{-ix})-\frac{e^{2ix}-e^{-2ix}}{2}+\frac{e^{3ix}-e^{-3ix}}{3}-\frac{e^{4ix}-e^{-4ix}}{4}+...$$

From the identity $e^{i n\theta}=cos n\theta+i sin n\theta$ we can derive $sin n\theta=\frac{-i}{2}(e^{i n\theta}-e^{-i n \theta})$, so the formula above becomes

$$ix=2i(\sin x-\frac{\sin 2x}{2}+\frac{\sin 3x}{3}-\frac{\sin 4x}{4}+...)$$

that is, $\frac{1}{2}x=sin x-\frac{sin 2x}{2}+\frac{sin 3x}{3}-\frac{sin 4x}{4}+...$

Notice what happened here: starting from a Taylor series, and using complex numbers as the bridge, we arrived at the Fourier series of a function. This can't help but come as quite a surprise and a delight, because in the real world the coefficients of the two are obtained through two seemingly completely different operations—differentiation and integration. If, in the complex world, they really are one and the same thing, then we're led to a striking conclusion:

Complex numbers reveal a hidden connection between differentiation and integration!

In fact, this is precisely at the heart of complex analysis! And in later posts, Taylor series and Fourier series will be recurring themes we come back to often.

The discussion above also suggests an idea: to obtain the Fourier series expansion of a function $f(x)$, replace the x inside with $x=-i[ln(e^{ix}+1)-ln(e^{-ix}+1)]$, turning it into the form $f(-i[ln(p+1)-ln(q+1)])$, and then expand it as a two-variable Taylor series in p and q. Since the coefficients of a Fourier series are computed via definite integrals, this seems to offer a shortcut method for computing definite integrals?? After all, differentiation is so much easier than integration... Seeing an example like this, all sorts of tempting ideas start bubbling up, and it's hard not to get excited. But, alas, things aren't quite so rosy. Our goal is to expand things as combinations of powers of $e^{ix}$ and $e^{-ix}$, but even a single term like $e^{ix}$ turns out to be a sum of infinitely many terms (any term of the form $p^{n+1}q^n$ is really a $e^{ix}$ term). This gets a bit fiddly to explain in words—readers who are interested should try working it out by hand, and things should become clear enough.