Functions

Why do functions have to be "one to one"? I don't understand the purpose of "not having two outputs for a single input".

There are several reasons for it: First of all, mathematics is mostly about precise definitions. What would you do if you had a function f which has the mapping 2 → 3 and 2 → 4?
Is f(2) = 3 or 4? If f(2) = 3, wouldn't that mean that 3 = 4 (because f(2) = f(2) and f(2) = 3, f(2) = 4)? If functions were a broader term and you had the possibility to have several mappings for one x to multiple ys, you couldn't use normal operations nor equality. And without equality, things look pretty grim in mathematics.

Basically, having many-to-one or many-to-many functions would require a pretty drastic redefinition of mathematics and would make it all more complicated.

If anyone knows:

Why is Γ(z)=(n-1)! considered the "best" representation of the factorial function, even though it's divergent at every negative integer? What does it mean to have a negative factorial anyways? Because of fractional calculus, should I be under the impression that (-n)!=1/n!? Why does Γ(z) suggest otherwise. Anyways, I'm just curious because I saw the other representations on this page. I don't have actual values for Hadamard's Gamma (and I couldn't find any), but from what it looks like, (-n)!=1/n! on Hadamard's Gamma.

Well, it's more generally defined as



It is not defined for non-negative numbers. Hadamard's Gamma does not have the same properties as Euler's Gamma. Euler's Gamma is the only function which is log-convex, has f(x) = 1 and f(x + 1) = xf(x). I assume you know of everything but the log-convex-part, so let's just take a short proof on that:

By the definition of the Euler-gamma, Γ(X) is positive for positive values of x. If you then let x, y > 0 and 0 ≤ λ ≤ 1, then



Thus, it is log-convex. And it's quite important, for example, in computer science, that it is log-convex. If it wasn't, one couldn't (in some cases) show lower and upper boundaries on the time complexity of a computer program/algorithm.

Noah