Why does being holomorphic imply so much about a function?

I haven't yet started my complex analysis course (soon!), but recently (inspired by you guys) I've been looking into holomorphic functions. And wow, they're cool! There's so much stuff that's true about them... But my question is: why is being holomorphic such a strong condition? Is there some intuitive reason why this seemingly simple condition implies so much about a function?

edit: may I add that I am thinking in particular about entire functions. e.g. it is not all obvious to me why being differentiable on the complex plane would imply Picard's theorem, that the function takes every value except at most one.

Solution 1:

I would contend that what makes complex analytic (holomorphic) functions so special is the structure of the complex numbers themselves. The fact that the complex numbers are essentially $\mathbb{R}^2$ and are also a field is a small miracle. This miracle is at the heart of the special behavior of complex analytic functions.

It is the two dimensional nature of the plane and the field multiplication allow us to see the C-R equations. My favorite way to see the C-R equations is to consider $$f'(z) =\lim_{\Delta z\to 0} \frac{f(z+\Delta z)-f(z)}{\Delta z} $$ Taking in turn $\Delta z =\Delta x$ and $\Delta z =i \Delta y$ the C-R equations just appear. This requires two dimensions for the two approaches and utilizes that $\mathbb{C}$ is a field when dividing by $\Delta z$. (Other chain rule type proofs of the C-R use both these facts, but are somewhat more subtle about it.)

Surely the Cauchy-Integral Formula, which tells us that a holomorphic function is in fact $C^\infty$ deserves mention, in fact it too is a consequence (less directly) of the two dimensional nature of $\mathbb{C}$ and field structure. The standard proof is to use Cauchy's Integral Theorem, which says that if $f$ is a holomorphic on a simply connected domain, then $$\int_\gamma f =0$$ for any sufficiently nice closed $\gamma$ curves in the domain. This itself is seen by applying Green's (2-d real structure again) and the C-R (field structure).

There are many, many more special and cool behaviors of holomorphic functions, but all of them seem to sit on the miracle of the two structures of the complex numbers. Just to advertise a few, Identity Principle (being 2-d connected is easier than being 1-d connected), zero counting theorems like Rouche's (line integrals sometimes have cool answers), the open mapping theorem (zero counting theorems are super cool), Liouville's Theorem (Cauchy integral theorem again), on and on and on.

Solution 2:

I think the nicest way to 'picture' this intuitively without having started your course is the 'amplitwist' concept coined by Tristan Needham (I'd recommend his book, Visual Complex Analysis, if this idea interests you).

Essentially, to quote from his exposition:

Analytic mappings are precisely those whose local effect is an amplitwist: all the infinitesimal complex numbers emanating from a single point are amplified and twisted the same amount.

This notion can be tightened up and made rigorous (see Needham), but it doesn't need rigour to provide illumination. Unfortunately I can't give you a direct intuition transplant, but I'd encourage you to play around a little with functions in $ \mathbf{C} $ and $ \mathbf{R}^2 $ to see if you can understand both what the above condition means, and why it might be significantly more restrictive than mere $ \mathbf{R}^2 $ analyticity.

So to directly answer some of your questions:

• Is there some intuitive reason why this seemingly simple condition implies so much about a function?

Yes -- and it's because it's not a simple condition! Essentially because of the required interplay between real and complex parts that is shown either algebraically in the C-R equations, or geometrically in the 'amplitwist' idea (there are amplitwist diagrams about online, and hundreds in Needham's book -- some might help).

• it is not all obvious to me why being differentiable on the complex plane would imply Picard's theorem

You're certainly not alone there. I think everyone is a little surprised by Picard's ('Great'!) theorem. I'm perpetually slightly surprised by it. But then, many of the major results in complex analysis are highly non-obvious. In some sense analytic functions are a major object of study precisely because their definition leads to a great number of 'nice' and interesting properties.