Tuesday, October 26, 2021

[xjcpfpwq] entire functions

nice are functions defined everywhere, continuous everywhere, and that have as many derivatives as you want (infinitely differentiable, smooth) everywhere.

complex functions (functions of a complex input yielding a complex output) that have this property are called "entire".

digression: what happens if we pay attention only to real?  i suspect things become more difficult.  if so, complex analysis paradoxically makes things simpler.  (wikipedia: smooth but not analytic, see also articles that link to it.  "smooth but not analytic" is not possible for complex functions, though i don't understand why.)

our interest in entire functions is motivated by wanting to keep things simple, studying the pretty things (first).

entire functions are a subset of holomorphic functions and a subset of meromorphic functions.

(digression: the distinction between holomorphic and meromorphic can be weird.  consider the following:

all holomorphic functions are meromorphic functions.  this is not controversial.  meromorphic functions are holomorphic functions with the additional flexibility of permitting poles, but you don't have to use that flexibility.

provocatively, all meromorphic functions are holomorphic functions.  to see any meromorphic function as a holomorphic function, define the holomorphic function's domain to be the meromorphic function's domain minus its poles.  this feels like cheating, but what distinguishes holomorphic from entire is that one can specify a domain less than the entire complex plane, so playing tricks with domain when defining holomorphic functions seems legal.

despite each category containing the other, they are not equivalent because we played tricks with the domain.  the meromorphic domain can be slightly larger (adding isolated poles) than the holomorphic domain.

end digression.)

even nicer than an entire function might be one that also does not go to infinity at infinity, staying finite even when given infinity as input.  all points on the complex plane infinitely far from the origin are equivalent, though i cannot explain why.  (this is not true for the real line: positive infinity is different from negative infinity.)  the domain "entire complex plane plus the 'point' at infinity" is the Riemann sphere.  unfortunately, the only entire functions which do not blow up at infinity are constant functions (consequence of Liouville's Theorem).  putting too many restrictions on beauty results in boringness.  it's surprising that it is not possible to construct an entire function that looks like a Gaussian bump or which oscillates everywhere like sin(x+y) in R^3.

digression: consider functions that are bounded at infinity but have exactly one singularity (so not entire) and are otherwise holomorphic everywhere.  i think the Riemann zeta function (with its single pole at s=1) is not in the class because it has unbounded sawtooth behavior at negative odd numbers.  functions in this class might always be entire functions transformed by a Moebius transform.  does it matter if the singularity is a pole or an essential singularity?

the Riemann xi function is entire, quite surprising since it is defined as a product of pieces (reciprocal, gamma, zeta), each of which has poles.

initially, i was worried that entire functions might be boring, perhaps only polynomials and variations around exp(z).  fortunately, this appears not to be the case.  there are many families of entire functions: see Wikipedia for examples.  how large and interesting is the set of entire functions?  this is subjective.  it appears research efforts are still in progress at characterizing them.  Weierstrauss factorization theorem is one important result.

entire functions are closed under composition.  also addition, multiplication, exp(f), integration, and differentiation.  raising to non-negative integral power.  (is raising to zeroth power tricky?  consider defining the result to be the constant function 1, ignoring 0^0.  but what if the input was the constant function 0?)  probably not other powers.

digression: how far can one go just with "polynomials and variations around exp(z)"?  probably quite far because we are including composition.  sin(x) = (1/2) * i * (exp(-i*x) - exp(i*x)) .  it seems the Weierstrass factorization theorem implies every entire function is a (possibly infinite) product of polynomials and variations around exp(z).

which entire functions can be computed quickly?  to high precision?  how?  some might require numerically solving a differential equation.

under what conditions does RootOf yield an entire function?  it might be rare.

how can we pick a random entire function out of a fairly large subset of transcendental (i.e., non-polynomial) entire functions?  everyone can have their own.  the Weierstrass sigma function offers one possibility.  for it, we need a way of randomly picking a lattice.  can Weierstrauss sigma be computed quickly?  the Weierstrass factorization theorem offers another possibility.  we need a way of randomly picking an infinite set of roots.  perhaps use a stream of random bits to choose a subset of a fixed ordered set of points on the complex plane.  we could also rotate a set of points by an integer multiple of an irrational angle.

entire functions are aesthetically the opposite of pathological functions.  nevertheless, how weird can entire functions be?

curiously, Weierstrauss's name is attached to an entire function (sigma, above) and to a pathological function, the Weierstrauss function, a fractal from a time before fractals became cool.

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