Anti-calculus proposition of Erdős

The “anti-calculus proposition” is a little result by Paul Erdős that contrasts functions of a real variable and functions of a complex variable.

A standard calculus result says the derivative of a function is zero where the function takes on its maximum. The anti-calculus proposition says that for analytic functions, the derivative cannot be zero at the maximum.

To be more precise, a differentiable real-valued function on a closed interval takes on its maximum where the derivative is zero or at one of the ends. It’s possible that the maximum occurs at one of the ends of the interval and the derivative is zero there.

The anti-calculus proposition says that the analogous situation cannot occur for functions of a complex variable. Suppose a function f is analytic on a closed disk and suppose that f is not constant. Then |f| must take on its maximum somewhere on the boundary by the maximum modulus theorem. Erdős’ anti-calculus proposition adds that at the point on the boundary where |f| takes on its maximum, the derivative cannot be zero.

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9 thoughts on “Anti-calculus proposition of Erdős

  1. That depends on how you extend the notion of differentiability to two variables. If by differentiable you mean partial derivatives exist, then yes, analytic is much stronger. But if by differentiable you mean the one-variable definition extended with complex numbers, then differentiable and analytic are equivalent.

  2. This proposition relies on the domain being a closed disk, rather than merely being the compact closure of a simply connected set (as for the maximum-modulus theorem which you quote).

    You can get a maximum at a vanishing-point of the derivative provided it’s not just at “one of the ends” (a boundary point) of the domain, but at a sufficiently sharp corner.

  3. Yeah, I think this is a much older result, possibly going back to Cauchy’s time. It’s directly related to the fact that solutions of Laplace’s equation cannot have local maxima.

    This later fact can be made sense of by noting that Soap films or rubber sheets stretched over a closed wire frame will take on a shape that is a solution to Laplace’s Equation. Any local maxima that wasn’t on the boundary (wire frame) would get stretched tight and disappear. A local maxima wouldn’t be a stable equilibrium in other words.

  4. There is certainly a confusion there. Do analytic functions – or just complex variables – ever attain maximum? Their modulus, of course, may. Also, for a real variable the derivative at a boundary point need not exist, let alone be zero.

  5. “A standard calculus result says the derivative of a function is zero where the function takes on its maximum. ”

    Are you sure about that? Isn’t it rather that the derivative is either zero or doesn’t exist where the function takes on a maximum? This latter aspect isn’t a trivial aspect as you know.

    I’m out of practice with my complex analysis these days though so I can’t comment on how this may or may not related to the picture in the complex domain.

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