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See also Emil Artin's elegant pamphlet, "The Gamma Function".
It's awesome and weird that there's a consistent and useful function that lets you define an equivalent to (-1/2)! -- and that what you get is the square root of pi.
Isn't it just a special case of the idea that the product of an empty sequence is 1?
Almost, but not quite. You could say "n! is the product of the natural numbers However, this same reasoning would say (-1)! = 1, (-2)! = 1, etc. And while you could define negative factorials that way, it's better in practice to leave negative factorials undefined.
In school it was explained through combinatorics, where the factorial among others tells, in how many ways you can order your set. And an empty set has just one, but it has one, order in which it can be arranged. Now that I am writing this, I'm losing confidence, that this explanation is mathematically correct..
That is the usual explanation, but I don't think that helps much since counting the number of ways to arrange nothing is confusing!
The number of ways to do A and then B is the number of ways to do A times the number of ways to do B. The number of ways to do nothing and then A should be the same as the number of ways to do A. This implies that the number of ways to do nothing is the multiplicative identity, one.
0! Is whatever makes sense in context, like any other defintion. Arguing about what it should be "in general" is treating math like reality television, and it makes you dumber. O! Isn't a big controversy, but the idiocy surrounding O^0 (which is usefully different in combinatorics vs in analysis) is mind-boggling.
• If p(x) = ∑[n=0..∞] a_n x^n is a power series, then p(0) = a_0 is its constant term, but it makes no sense to write this unless 0^0 = 1.
• The power rule d/dx [x^n] = n x^(n−1) holds for n = 1 and x = 0, but this requires 0^0 = 1.
The reason that some people believe it’s important to undefine 0^0 in analysis is that the limiting expression
lim[x → a] f(x)^g(x)
does not necessarily exist when lim[x → a] f(x) = lim[x → a] g(x) = 0. But all this means is that we have to draw a distinction between the _value_ 0^0, which equals 1, and the indeterminate _limiting form_ 0^0, which is an abbreviation for the above type of limit.
It is not uncommon for values to evaluate differently from the corresponding limiting forms. For example, the value floor(0) equals 0, but the limiting form floor(0) is indeterminate. It may seem surprising that such a discrepancy arises for exponentiation, but all it means is that exponentiation is discontinuous at (0, 0), as it must be.
(Note however that the above limit _does_ exist with mild conditions on f and g: if f, g are complex analytic functions with f not identically zero, then the limit equals 1.)
See also Donald Knuth’s _Two notes on notation_: http://arxiv.org/abs/math/9205211.
0! Is whatever makes sense in context, like any other defintion. Arguing about what it should be "in general" is treating math like reality television, and it makes you dumber. O! Isn't a big controversy, but the idiocy surrounding O^0 (which is usefully different in combinatorics vs in analysis) is mind-boggling.
Defining 0^0 = 1 is just as important in analysis as it is in combinatorics, even though analysis texts aren’t always clear that they’re making use of this fact. • If p(x) = ∑[n=0..∞] a_n x^n is a power series, then p(0) = a_0 is its constant term, but it makes no sense to write this unless 0^0 = 1. • The power rule d/dx [x^n] = n x^(n−1) holds for n = 1 and x = 0, but this requires 0^0 = 1. The reason that some peop…
Earlier quoted context omitted.
Defining 0^0 = 1 is just as important in analysis as it is in combinatorics, even though analysis texts aren’t always clear that they’re making use of this fact. • If p(x) = ∑[n=0..∞] a_n x^n is a power series, then p(0) = a_0 is its constant term, but it makes no sense to write this unless 0^0 = 1. • The power rule d/dx [x^n] = n x^(n−1) holds for n = 1 and x = 0, but this requires 0^0 = 1. The reason that some peop…
Defining 0^0=1 has issues. Example: a(n)=1/exp(n), b(n)=1/n. Then both a and b converge to 0, but a^b is always exp(-1) (and hence does not converge to 1).
Similarly, floor(−1/n) converges to −1, which tells us that the _limiting form_ floor(0) is not always equal to 0; but the _value_ floor(0) is still equal to 0. Nobody uses this to argue that the value of floor(0) should be undefined or context-dependent.
See also Emil Artin's elegant pamphlet, "The Gamma Function".
It's awesome and weird that there's a consistent and useful function that lets you define an equivalent to (-1/2)! -- and that what you get is the square root of pi.
Earlier quoted context omitted.
Defining 0^0=1 has issues. Example: a(n)=1/exp(n), b(n)=1/n. Then both a and b converge to 0, but a^b is always exp(-1) (and hence does not converge to 1).
What you have shown is that the _limiting form_ 0^0 is not always equal to 1. The _value_ 0^0 is still equal to 1. Similarly, floor(−1/n) converges to −1, which tells us that the _limiting form_ floor(0) is not always equal to 0; but the _value_ floor(0) is still equal to 0. Nobody uses this to argue that the value of floor(0) should be undefined or context-dependent.
But you are right, what I meant is there is no way to define 0^0 maintaining continuity of the power function. Why is this important? Because power is a continuous function otherwise.