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In Ahlfors' Complex Analysis he remarks that harmonic $u(x,y)$ can be expressed as $$ u(x,y) = \frac{1}{2}[f(x + i y) + \overline{f}(x - i y)] $$ when $x$ and $y$ are real. He then writes

"It is reasonable to expect that this is a formal identity, and then it holds even when x and y are complex".

What does he mean in this context by "formal identity"?

Edit: This entire page (p.27 of my edition) comes with what is a caveat, as far as I can tell:

We present this procedure with an explicit warning that it is purely formal and does not possess any power of proof.

In the same page he uses the phrases "formal procedure", "formal reasoning", "formal arguments", and "formal identity".

Is he more or less saying that he's embarking on something that could be considered suspect, at least at this point in the book?

Thank you very much!

bryanj
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    I think that "formal" here refers to the "form" of the identity's equation rather than the "formality" or "rigorousness" of it. That is, he means that the identity will be an equation of the same form regardless of whether $x$ and $y$ are allowed to be complex. That's my guess, anyway. – Ben Grossmann Jul 16 '13 at 23:37
  • Harmonic functions are smooth with convergent power series. He might be referring to the identity between the (formal) power series. After all, looking at the power series is a way to proving the claim. – OR. Jul 16 '13 at 23:40

1 Answers1

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The word "formal" as it's being used here doesn't have an entirely rigorous meaning. The archetypal example of a formal argument is manipulating a power series without worrying about convergence, which gives rise to the notion of formal power series. In general, a formal argument is one based on the "form" of the mathematical objects involved without thinking about their "substance" (e.g. a power series is a form, a function it's a Taylor series of is a substance).

In this case I agree with RGB that a possible interpretation is that the identity might hold on the level of power series, in which case it should hold for even complex $x, y$.

Keenan Kidwell
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Qiaochu Yuan
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