On our most recent episode of My Favorite Theorem, my cohost Kevin Knudson and I were happy to talk with Steven Strogatz, an applied mathematician at Cornell and author of several popular mathematics books. You can listen to the episode here or at kpknudson.com, where there is also a transcript.
Strogatz told us about the Cauchy integral theorem, also known as the Cauchy–Goursat theorem, in complex analysis. As longtime readers of a blog called Roots of Unity are probably aware, complex does not mean complicated in this context. It refers to complex numbers, numbers of the form a+bi, where a and b are real numbers and i is defined to be the square root of −1. You can think of the complex numbers as the x-y plane with a little bit of extra structure. Just like with the real numbers, we can add, subtract, multiply, and divide complex numbers. We can define complex functions, which take complex numbers as their inputs and produce complex numbers as their outputs, which also allows us to do all the fun things we like to do with real functions.
The Cauchy integral theorem is a nice theorem related to how functions in the complex plane behave, and I especially enjoyed hearing about Strogatz’s personal reaction to the theorem when he first learned it. But I thought something else we talked about in the episode was even more interesting: What’s so great about complex analysis?
Complex analysis is basically calculus in the complex plane. Calculus, the study of continuous change, is one of our favorite things to do with real functions. The field of analysis expands calculus from the realm of functions of one variable to more complicated domains and multivariable functions. Naively, you might assume that if the complex plane is just a two-dimensional plane with one extra rule (i2=−1), single-variable complex analysis would be similar in flavor to two-variable real analysis.
Most mathematicians don’t feel that way. Complex analysis feels pristine and perfect, while real analysis seems messy and wild. Real analysis is full of “monstrous” objects like the Weierstrass function Ben Orlin talked about a few months ago, but in complex analysis, everything just works out. For example, if a function is differentiable once in the sense of complex analysis, it's infinitely differentiable. Being a little bit smooth means a function is infinitely smooth. On the other hand, real functions can be differentiable 17 times and then break on the 18th. There are no guarantees.
I love complex analysis, almost to the point of reverence. But while we were talking, I started thinking about why complex analysis seems so much more perfect than real analysis. It all comes down to what functions we let in the door. Theorems like the Cauchy integral theorem apply only to functions that are complex analytic, which means their derivatives obey restrictive rules about how they interact. The equivalent notion in real analysis allows for much more disorderly functions. The theorems about complex analytic functions apply to a narrower swath of functions. It’s like comparing Beyoncé’s backup dancers to everyone who’s ever had a solo dance party in their kitchen. A choreographer could do a lot more with the former than the latter. Likewise, a mathematician can prove a lot more powerful theorems with complex analytic functions that with real analytic functions.
I’m a little torn: Is complex analysis beautiful because its functions are almost miraculously well-behaved, or is it shallow because it applies to such a restricted set of functions? I almost talked myself into believing the latter, but then I came back around to the source of the difference between complex and real analysis: i. The powerful, almost-too-good-to-be-true theorems of complex analysis are all consequences of defining a number i to be the square root of −1. The rules of multiplication in the complex plane fall naturally out of that definition, and the rest of the calculus follows. And I think I have to admit that the power of that one rule to create an entirely distinct discipline is beautiful to me.