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.

Answer

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 R2 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 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.

Attribution
Source : Link , Question Author : ShakesBeer , Answer Author : Joe Manlove

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