Liouville's theorem
/ lyoo-VEEL /
An entire function is one that is holomorphic on the whole plane — no singularities anywhere. Liouville's theorem says such a function has nowhere to hide: if it is also bounded (its modulus stays below some fixed number everywhere), then it must be constant. A holomorphic function defined on all of the plane simply cannot stay small without being flat.
The proof is a one-line consequence of the Cauchy estimate for the first derivative. Pick any point z_0 and any radius R. The estimate gives |f'(z_0)| at most M / R, where M is the global bound on |f|. Since f is entire, R can be made arbitrarily large, so M / R can be pushed below any positive number; therefore f'(z_0) = 0. As z_0 was arbitrary, f' vanishes identically, and a function with zero derivative on a connected region is constant. Boundedness plus entirety forces zero slope everywhere.
This is the rigidity of holomorphy at its most dramatic, and it has teeth. It immediately yields the fundamental theorem of algebra. It also explains why so many interesting entire functions, like e^z, sin z, and polynomials, are necessarily unbounded — there is no nonconstant bounded one to be found. A useful sharper cousin: if an entire function grows no faster than a polynomial of degree d, it must itself be a polynomial of degree at most d.
The function sin z is entire but unbounded — for purely imaginary z = i y, sin(i y) = i sinh(y) blows up as y grows — exactly as Liouville's theorem requires, since a bounded entire sin would have to be constant.
Why complex sine, unlike real sine, cannot stay bounded.
Boundedness on a part of the plane is not enough; the theorem needs the function to be entire (holomorphic everywhere) and bounded everywhere — a function bounded only on a disk can certainly be nonconstant.