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:
“A leaf that is supposed to grow is full of wrinkles and creases before it develops; if one doesn’t have the patience and wants the leaf to be as smooth as a willow leaf from the start, then there is a problem.”
—Johann Wolfgang Von Goethe (1749–1832)
“As a medium of exchange,... worrying regulates intimacy, and it is often an appropriate response to ordinary demands that begin to feel excessive. But from a modernized Freudian view, worrying—as a reflex response to demand—never puts the self or the objects of its interest into question, and that is precisely its function in psychic life. It domesticates self-doubt.”
—Adam Phillips, British child psychoanalyst. “Worrying and Its Discontents,” in On Kissing, Tickling, and Being Bored, p. 58, Harvard University Press (1993)
“But as to women, who can penetrate
The real sufferings of their she condition?
Man’s very sympathy with their estate
Has much of selfishness and more suspicion.
Their love, their virtue, beauty, education,
But form good housekeepers, to breed a nation.”
—George Gordon Noel Byron (1788–1824)
““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)