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Is there an approach to complex analysis that is fundamentally different from the usual route via Cauchy's theorem? For example, can one prove that a complex-differentiable function is given locally by its Taylor series 'directly' from the Cauchy-Riemann equations (and ideally in a reasonably elementary way, i.e. using not much more machinery than is needed for the standard method)?

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Harmonic functions share many of the properties you also have for analytic functions (see section properties of harmonic functions of the Wikipedia article for harmonic functions). Because any function fulfilling the Cauchy-Riemann equations is harmonic, you can have a look at the proofs for the properties of harmonic functions to have an alternative way to prove those theorems of complex analysis.

Unfortunately not each theorem can be proved in this way. The Identity theorem will be missing for example (which is really important for complex analysis). But on the contrary harmonic functions are also defined on real valued vector spaces with dimension higher than 2.

Stephan Kulla
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  • Thanks @tampis. You say the identity theorem will be missing, but it sounds (from some Googling) like harmonic functions are real analytic. Is this not enough to deduce the identity theorem? –  Jul 25 '14 at 11:33
  • @Jez: See http://math.stackexchange.com/questions/156505/uniqueness-theorem-for-harmonic-function for an answer... – Stephan Kulla Jul 25 '14 at 23:22
  • Ah, I think we meant slightly different things! I was thinking of the identity theorem as "if a function is zero on an open set then it's zero on the whole component", whereas it looks like you meant "if the zero set has an accumulation point then the function is zero on the whole component". So I guess we were both sort of right. Thanks again @tampis. –  Jul 27 '14 at 09:20