扩展：什么是复分析?

If you'd notice, everything covered about power series covered previously concerns real polynomials and real values. When we wrote a series, it was as an infinite polynomial with real coefficients.

There's just one problem. Real numbers aren't the only set of numbers.

Also, this part of calculus has pretty much been completely studied since Brook Taylor first explained Taylor series 300 years ago (we've just been rigorizing his work).

The obvious extension then, is complex analysis. It sounds mysterious, but personally, I feel it is one of the most beautiful areas of mathematics. And to get started, let's just plug a complex number, $i\cdot y$, into the Taylor series for $e^x$.

We know $$e^x = \frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$ We previously assumed $x$ was real, but let's just plug in the complex number $iy$ in for $x$. Note we no way of knowing this works. Mathematicians like Leonhard Euler and Oliver Heaviside had no way of knowing either, but they did it anyway and waited for later mathematicians to justify their work. We'll get $$e^{iy} = \frac{1}{0!} + \frac{iy}{1!} + \frac{(iy)^2}{2!} + \frac{(iy)^3}{3!} + \frac{(iy)^4}{4!} - \cdots$$ Figuring that can be cleaned up, we take out all the powers of $i$ and see $$e^{iy} = \frac{1}{0!} + i\frac{y}{1!} - \frac{y^2}{2!} - i\frac{y^3}{3!} + \frac{y^4}{4!} + \cdots$$ Surprisingly enough, we can get some useful information out of this (compare the usefulness gained from substituting $x=-y$ into $e^x$)! If you have a keen eye, you'll note the above implies $$e^{iy} = \left( \frac{1}{0!} - \frac{y^2}{2!} + \frac{y^4}{4!} + \cdots \right) + i \left( \frac{y}{1!} - \frac{y^3}{3!} + \cdots \right) = \cos(y) + i\sin(y)$$ even though we haven't defined complex exponentiation and this makes no sense. However, this must be true, because it's Euler's formula! If you let $y = \pi$, you'll even understand the origin of Expii's name*.

Ok, so we've substituted imaginary numbers into Taylor series. Fair enough. The natural question is to ask for the radius of convergence of the above series for $e^{ix}$.

Let's just use the typical power series test. The $n$th term of the Taylor series for $e^{iy}$ is $\frac{(iy)^n}{n!}$. Note $$\lim_{y\rightarrow \infty} \left| \frac{(iy)^{n+1}/(n+1)!}{(iy)^n/n!} \right| = \lim_{y\rightarrow \infty} \left| \frac{iy}{n+1} \right| = 0 < 1$$ which means this series definitely converges, no matter what you believe about $e^{iy}$.

Speaking of convergence, why exactly is it called the radius of converge? Don't ask me. I just want to find the radius of converge of the following series: $$\frac{1}{1-z} = 1 + z + z^2 + z^3 + \cdots$$ where $z$ is a random complex number. The power series test tells us to find $$\lim_{z\rightarrow \infty} \left| \frac{z^{n+1}}{z^n} \right| = |z|$$ which must be less than $1$ for this series to absolutely converge. But wait, the graph of $|z| = 1$ looks like this:

and the graph of $|z| < 1$ is just that circle shaded in, formally called a disk. It should now be sort of clear why we refer to $|z| < 1$ as the "radius of convergence"--it literally represents the radius inside which the series converges.

That's basically it for a very elementary introduction to complex analysis. Of course, you may noticed I breezed over something: when I wrote the limit, I wrote $z \rightarrow \infty$. Except, you may have noticed $z$ can go in an infinite number of directions...which one is infinity? If we refer to the standard notion of $+\infty$, that still leaves an infinite number of directions for $z$ to grow in. In the world of real numbers, numbers can only tend to infinity in the $+\infty$ or $-\infty$. In the world of complex numbers, there's also $+i\infty$, $-i\infty$, and an infinite number of directions in between. There's no way we can formalize limits in this way! By the way, derivatives also face the same problem--what the heck is a $dz$? A tiny distance from $z$ in what direction?

There's a way around this, and if you go into this field, or just take a vector calculus course, you'll learn that the derivative of the function $f(x+iy) = f(z)$ is the crazy expression (with partial derivatives) $$\frac{\partial f}{\partial x} \frac{\partial x}{ \partial z} + \frac{\partial f}{\partial y} \frac{\partial y}{ \partial z}$$ which comes about as a result of one of an infinite number of chain rules (yep, plural), and with the basically vector-valued function $z$ having the derivative $dx + idy$.

Complex functions, in turn, are governed by the innocuous-looking Cauchy-Riemann equations: $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$ $$\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}$$ where $u$ and $v$ are functions of the real coefficients $x$ and $y$.

All of that might have been mathematical nonsense. That's okay. It'll make sense if decided to pursue the field of study that lets you find $\arcsin(e)$ and show that $i^i$ somehow is equal to an infinite number of reals, but not really (aka, complex analysis). But more importantly, for the longest time, everything above made no sense. But with the random exploits of a few visionaries, and the work of hundreds of mathematicians formalizing all these arguments, complex analysis was slowly molded into a field with applications as diverse as number theory and electrical engineering. It might seem difficult, because it is (it's still active, for one), but this topic is in and of itself rewarding.

*By the way, the fact that $e^x = 1 + x + \frac{x^2}{2} + \cdots$ helps us in significantly more ways. It turns out you can let $x$ be nearly anything, and you'll create whole new areas of mathematics from it. For example, I'll leave you with these two equations: $$e^A = I \text{ for the matrix } A = \text{the 0-matrix}$$ $$e^D f(x) = f(x+1) \text{ where } D \text{ is the derivative operator}$$ Those are both true, but...