4 ms·
The miracle of complex numbers is that they have two seemingly unrelated but very nice properties: 1) Every non-constant polynomial has a root 2) Every holomo
by cousin_it 9y ago
The miracle of complex numbers is that they have two seemingly unrelated but very nice properties:
1) Every non-constant polynomial has a root
2) Every holomorphic function is analytic
The only connection I know is that (2) implies Liouville's theorem which implies (1), but it doesn't seem any less miraculous.
- mfukar 9y agoNot miraculous at all. The motivation for complex numbers came out of trying to solve cubic polynomials (Tartaglia, Cardano).
- mbid 9y agoWhat does that have to do with every function being analytic? Anyway, you can solve every cubic polynomial in the algebraic closure of Q, which is far less than C, so you don't need to construct the complex numbers for that.
- contravariant 9y agoThe closest link I found from (1) to (2) uses differential geometry and in particular the exterior algebra. Using the exterior derivative df = (df/dx) dx + (df/dy) dy. it is possibly to define (identifying C with R^2) a function to be holomorphic when it can be written df = g (dx + idy) for some function g. From the properties of the exterior derivative you then get 0 = d(df) = d( g dz) = df' ^ dz, which implies that dg = h dz for some h, hence g is holomorphic as well. Defining the complex derivative of 'f' to be 'g' at least shows that any holomorphic function is infinitely complex differentiable. I haven't quite managed to figure out why this definition of the complex derivative coincides with the limit of f(w+z)/z as z->0 with z complex, but it should follow from the algebraic properties of C. Showing holomorphic functions to be analytic is somewhat trickier, but you can probably use that the above definition of holomorphicity and Stokes' theorem together imply Cauchy's integral theorem (every integral around a loop vanishes).