A Smooth Function Which Is Nowhere Real Analytic
A more pathological example, of an infinitely differentiable function which is not analytic at any point can be constructed by means of a Fourier series as follows. Let A:={2n : n ∈ N } be the set of all powers of 2, and define for all x ∈ R
Since the series converge for all n ∈ N, this function is easily seen to be of class C∞, by a standard inductive application of the Weierstrass M-test, and of the theorem of limit under the sign of derivative. Moreover, for any dyadic rational multiple of π, that is x:=π p/q with p ∈ N and q ∈ A, and for all order of derivation n ∈ A, n ≥ 4 and n > q we have
where we used the fact that cos(kx)=1 for all k > q. As a consequence, at any such x ∈ R
so that the radius of convergence of the Taylor series of f at x is 0 by the Cauchy-Hadamard formula . Since the set of analyticity of a function is an open set, and since dyadic rationals are dense, we conclude that f is nowhere analytic in R.
Read more about this topic: Non-analytic Smooth Function
Famous quotes containing the words smooth, function, real and/or analytic:
“I have seen her smooth as a cheek.
I have seen her easy,
doing her business,
lapping in.
I have seen her rolling her hoops of blue.
I have seen her tear the land off.”
—Anne Sexton (1928–1974)
“Of all the inhabitants of the inferno, none but Lucifer knows that hell is hell, and the secret function of purgatory is to make of heaven an effective reality.”
—Arnold Bennett (1867–1931)
“That is the real pivot of all bourgeois consciousness in all countries: fear and hate of the instinctive, intuitional, procreative body in man or woman. But of course this fear and hate had to take on a righteous appearance, so it became moral, said that the instincts, intuitions and all the activities of the procreative body were evil, and promised a reward for their suppression. That is the great clue to bourgeois psychology: the reward business.”
—D.H. (David Herbert)
““You, that have not lived in thought but deed,
Can have the purity of a natural force,
But I, whose virtues are the definitions
Of the analytic mind, can neither close
The eye of the mind nor keep my tongue from speech.””
—William Butler Yeats (1865–1939)