the theorem's not yet proven, but that's not important. what is important is that i have a good idea and after thinking about it carefully, i think it will work. i think i can prove the theorem by tomorrow morning.
i'm starting to believe that the analysis gods accepted sacrifices in the form of discarded pages of scrawls. i think i've used about 10-15 blank pages over the course of two or three days -- front and back -- and i've thrown out two pens ..
.. well, maybe the pens aren't so relevant. i could also credit hard work, but as a friend of mine predicted, the solution is reasonably straightforward .. as always. in fact, an educated guess would have led straight to the methods of proof.
so let me not credit any hard work; instead i should credit and discredit my stubbornness of habit.
first, i should swallow my pride.
in a previous post i dismissed the use of computation to arrive at results about harmonic functions, in favor of more elegant and geometric means to tackle the problem. so i thought of many ideas and not one of them worked, and quickly i wondered if there was enough data in the problem to warrant any reasonable answer.
in the end, the optimistic outlook gave me the current idea. i stopped worrying about formality and rigor, took limits and guessed where they should go. if, for some reason, the quantities converged to what i hoped they would, then i'd at least obtain a foothold and could climb through a proof.
to understand the convergence, i had to compute .. and compute .. and compute. now i think i have some notion of why it all works, without appealing to computations, but the notion would not have appeared had i not computed. it reminds me of something (similar) that a certain prof @ syracuse told me (and the rest of the audience at his talk):
one must be very careful and very patient with computations, but if you are careful then they will teach you morals.
i suppose that prof was right .. at least about my last few days.
anyways, the proof is incomplete. there's work to be done, and teaching preparations to meet. tomorrow shall be a busy, busy day.
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