Difference between revisions of "06-1350/Some Equations by Kurlin"

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** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
 
** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
 
** If there are further ones, why haven't we heard about them? Or else, where do we read about them?
 
** If there are further ones, why haven't we heard about them? Or else, where do we read about them?
** If there aren't further ones, this is a lovely "closure" property  
+
** If there aren't further ones, this is a lovely "closure" property enjoyed by exponentials. How come I don't know it yet?
 +
 
 +
* Assuming not all tame equations have tame solutions, what was special about our equation, that lead to the existance of a tame solution? Can you write a very general condition on tame equations that will ensure the existance of a tame solution?
  
 
{{note|Kurlin}} ''Compressed Drinfel'd Associators,'' V. Kurlin, {{arXiv|math.GT/0408398}}.
 
{{note|Kurlin}} ''Compressed Drinfel'd Associators,'' V. Kurlin, {{arXiv|math.GT/0408398}}.

Revision as of 18:56, 17 September 2006

Claim [Kurlin]. The function f(\lambda,\mu) defined by

1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right)

Satisfies

f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right).

If this class is successful, at some point in the future I will tell you why I care. But meanwhile a few questions to expose my ignorance of simple algebra. Note that f is is a rational function in its variables and their exponentials. Let's call such functions "tame". The equation f satisfies involves rational expressions in the variables and their exponentials and linear substitutions applied to f. Let us call such equations "tame". (The equation is also linear in f. You may or may not include this in the definition of "tame"). So we see that a tame equation has a tame solution.

  • Is that always the case?
    • A prety example is the tame equation e(x+y)=e(x)e(y), whose solution is the "first" transcendental function e^x.
    • Are we done defining "new" special functions after we've defined e^x or are there further ones that cannot be defined in terms of it?
    • If there are further ones, why haven't we heard about them? Or else, where do we read about them?
    • If there aren't further ones, this is a lovely "closure" property enjoyed by exponentials. How come I don't know it yet?
  • Assuming not all tame equations have tame solutions, what was special about our equation, that lead to the existance of a tame solution? Can you write a very general condition on tame equations that will ensure the existance of a tame solution?

[Kurlin] ^  Compressed Drinfel'd Associators, V. Kurlin, arXiv:math.GT/0408398.