Question

The following is a theorem proven on page 127 of Ahlfors: If f is holomorphic in a (connected) region Ω⊆C and, for some a∈Ω,f (n) (a)=0 for all integers n≥0, then f is identically zero in Ω. (a) Show that the same is not true if you replace C with R and consider infinitely differentiable functions f:R→R. Hint: Use f(x)=e −1/x 2 (extended to 0 at x=0) and a=0. For fun, plot this function if you are not sure what it looks like. (b) Why does the same function f(z)=e −1/z 2 (extended to 0 at z=0 ) not give a counterexample to the theorem in the original setting?

Answer 1

The theorem does not hold for infinitely differentiable functions on the real line. An example is f(x) = e^(-1/x^2), extended to 0 at x = 0. This function has all derivatives equal to zero at x = 0 but is not identically zero on the entire real line.

Step-by-step explanation:

The theorem states that if a holomorphic function is identically zero at all orders of derivatives at a point in a connected region in the complex plane, then the function is identically zero in the entire region. However, this is not true for infinitely differentiable functions on the real line. An example of such a function is f(x) = e^(-1/x^2), extended to 0 at x = 0. This function is infinitely differentiable and has all derivatives equal to zero at x = 0, but it is not identically zero on the entire real line.

The reason this function does not give a counterexample in the complex plane is because it has an essential singularity at z = 0, which prevents it from being extended smoothly to z = 0. In other words, the behavior of the function near z = 0 is different in the complex plane compared to the real line.