Let's discuss Mathematics

not sure if I expressed it very well

So this is the actual expression I'm looking at, so instead of x it's T which is actually inverse of it sort of



So high T I can work with (analogous to x in the limit of zero)
but low T I'm not so sure, because then the T² goes to zero, but the exp goes to infinity
Actually it's kinda worse than that because T doesn't go to infinity, just something very big

It is a physics problem, and then it's not always like "proper math" going on

like in a previous problem in the same problem sheet, I got an answer for high T that was like hw/2+kT and simplified that to just kT, since that term dominates. It's really more stuff like that I kinda want to get

 

Taylor expansion gets you to the end most of the time with these kinds of functions. exp(hw/kT) = 1 + hw/kT + ((hw/kT)^2)/2 + ((hw/kT)^3)/6 + etc... You then substract 1 and deal with the square in the denominator. Be careful about which term is actually the "strongest" because the parts of the sum are going to infinity when T gets lower.

 

Looks similar to Planck's law, at least in the general form of the equation. It has two approximations for both short and long wavelengths, Rayleigh-Jeans and Wien respectively. Maybe looking up how you get from the general case to the long and short approximations would help.

 

Hi, I have a general question:
why specifically use the diagonal method to show that some sets have larger cardinality than that of the set of natural numbers?

I have to assume that the point was to show that one cannot order the sequence in any way which would correspond to a tie to the natural numbers, eg 1.1, 1.2 etc would still be placed in positions (eg) 1 and 2, but then the full set would take up all of the positions used by natural numbers and still allow room for many more positions for the fractions. But why should one use a diagonal to show there are many more possible positions, instead of any other method? Is this the simplest possible to think of, or does it have any specific use in other things? I mean intuitive it would be self-evident that even a fraction of a fraction of a fraction... of something would go on in a one to one tie to the natural numbers, so is there some use in coming up with a specific and easy to iterate set which can be fed back to the original (different in the diagonal) and still allow for the new set having space for more?

To me the diagonal presentation seemed somewhat similar to the first proof of there being more prime numbers than can be accounted for, so was wondering if it was just one possible way of making the argument or something inherently valuable due to ties to other parts of set theory.

 

arccot(phi)=arctan(1/phi), yea?