harmonic functions are heart-breakers.

of all the various sorts of functions that i've studied lately, harmonic functions are the ones that break my heart.

they have so many wonderful properties, including

1. they are classically (C^&infin-) smooth,
   
2. they have the mean-value property,
   
3. harmonic extensions of homeomorphisms of circles are actually homeomorphisms of discs (the Rado-Kneser-Choquet Theorem),
   

yet it seems practically impossible to discern any information about inverses of these harmonic homeomorphisms. perhaps one may say something abstractly about them, but when one tries an actual computation ..

.. forget it. they will break your heart, every single time.

i can't say it's their fault [1]. if anything, the problem seems to be an incongruency between tools in topology and in measure theory, and "natural" approaches in one area are wholly unnatural in another.

.. bah. humbug. back to work, i suppose, or at least some semblance of what work should look like.

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[1] for the record, it also makes for inappropriate personification.