matt grime

What makes you think that pointwise limit is ever defined? At last you're using limits, but the point is why do you want a "function" on R that isn;t defined anywhere? this isn't even a fucntion, you know since it has no domain or range. that is why i am queasy - you are waving divergent objects around and completely misusng the word function

how about sqrt(-|x|)... but this is all very very bad mathematics. I really cannot emphasize that enough. so bad in fact that i believe the entire thread ought to be removed. sorry, but it is just a horrible abuse of the subject.

to be honest this horrible abuse of divergent products and improperly defined functions is making me queasy, can we do something more mathematical please?

you can't put the infinities in at the end! there are several such functions, indeed uncountably many. why not take the product over all rational numbers, suitably ordered, or just define it to be 1 at each rational and not defined at each irrational? you need to stop thinking of functions as things that have a nice formula in terms of x.

no - they aren't really special since it is a badly defined function of a real variable (ie not defined at all the points you are claiming are in the domain), and not an interesting one of a complex variable - at least that is my opinion.

you don't need to use log, and you are free to pick a different branch of each root for your fucntion - i was explaining why my answer was different from yours ie why my answer didn't give a real root to (-2)^{1/5}. I was simply picking the first root anticlockwise round the origin from the real axis (the principal branch of log) and using it if it was real and ignoring it otherwise. you were picking one that is real if possible and ignoring the rest. however, how would you decide what something to the power sqrt(3) was? it isn't important, though, but might explain where the log is useful. apologies for misquoting you.

As a matter of english infinity is a noun, not an adjective - you mean "infinite number of gaps" "any x can be expressed as a ratio..." no, there are irrational numbers too you know. the second part would depend upon whjat bracnch of log you were using - taking powers like 1/n gives n answers - you need to pick one ie pick a branch. the branch i picked is a uniqe branch - you are changing it for 1/3 and 1/5 and so on.

i think at least 10% of the threads i respond to could be answered by a simple: you asked about FOO, try googling for FOO+Wolfram.

As ever a simple definition can be obtained by looking on wolfram http://mathworld.wolfram.com/ClosedInterval.html

no, that product makes no sense: remember that an infinity appearing in an index of in a sum or product merely tells you to no stop at any finite point, and it cannot appear in the terms of the product as it is not a real number. What is an "infinity gap"? k^x is the same as exp{xlogk} log of minus 2 is log(2)+ipi, so you get a real number when xpi is an even integer, the "gaps" are just the complex answers that cannot be plotted on a real curve.

forget the wikipedia thing - i misread you and can't decide what the point is anymore.

Of course not, as Goedel proved over 80 years ago.

Nothing in mathematics is absolutely true. The axiom is true in ZFC but independent of ZF. If it's true then every vector space has a basis, but then the Banach-Tarski paradox must hold.

eh? you declared that sqrt should be positive - that means you made a choice, something that if you were really defining x^y properly would require you to comment on. i don't really care for that attitude towards acceptable ignorance of standard terms, and it isn't my fault you didn't learn them; if you wanted sympathy, or for me to care at all, then you shouldn't make such absolute claims about things in mathematics (ie what the product symbol is) only to reveal that you know none of the basics; problems you fidn are not ones in mathematics but ones in your knowledge. there is no absolute definition of principal - note correction to spelling. the principal block contains the trivial module the principal branch of log takes +ve reals to reals, principal divisors are something else entirely, a principal ideal is generated by one element.

principle simply means "the one that makes most sense and is most widely used" we could declare the square root function to be never positive so that ^2 and ^{1/2} are mutually inverse bijections between R^- and R^+ the negative and positive reals. but generally it is better to make them inverse bijections on R^+ to itself

when you state that sqrt(4)=2 you have chosen a branch of the square root function - the principle one.

there are no problems with i, and don't pretend there are.

no, what about the square root of 4? is it 2 or -2? no complex variables at all

you define x^y as exp(ylogx) since it is correct and negates any need to muck around with thinking what x^pi, say, as the limit of the rational approximations to pi - does that even exist? is the limit well defined? x^y is just a function, and functionally exp(ylogx) is equivalent to x^y and avoids the need to take a completion of the integral and hence rational powers to the real powers, and is generally the easiest definition. since it is obviuosly coninuous and henc, must be equal to the competion if it exists. Ie i am demonstrating that the completion is well defined by giving a function in the completed space that agrees at the rational points. you did omit branches when you claimed to definethe power 1/3 or 1/7.

You wrote [math]\prod_{k=1}^{y}x = x_1x_2\ldots x_y[/math] that is a false statement - the LHS and the RHS are not equal. There is nothing in your post that states what x_i is at all we are forced to introduce what looks like y new variables. incidentally, x^y is defined as exp{ylogx} defined for all x strictly positive, and even negative y picking a branch of log. Branches being yet one more thing you forgot in your attempt to define powers. In anyecase, none if this has indicted that you're explaining anything about 0^0 that is a genuine "funky" problem that we are glossing over in mathematics

we are doing no such thing since the produvt sign tells us already no, you are iontroducing a deduction that the reader will guess x_i=x for all i. there is no har in saying x.x...x n factors. and it avoids writing something that is as it stands false

I thikn you'll find that we all know what x^y means but we're trying to educate you in how mathematics works. There is no known consisitent with all uses way of defining the symbol 0^0. So? who said there was? However, x^0 has a removable singularity in a loose sense at x=0 which is the convention used in Taylor series

johnny, you are, again, missing the point. Who states that x^0 must be defined for any x, or that it must follow that x^nx^m=x^{n+m} if n+m=0? It is clearly true from the definitions if n+m=/=0 (and x=/=0 as necessary). And from that reasoning you can only demonstrate that x^0=1 makes sense , ie is consisitent, which is what we're trying to do here, for x=/=0. That last choice of "defined" you ocmment on was a bad one - better to say "exists" or perhaps "can be defined" if it can be defined unambiguously and if it follows the other rules of indices, then x^0=1 for x=/=0 And there should be no suffices on the right since there were none on the left in the product. But then you don't understand what how the product symbol works.