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:
“Yet I’ll not shed her blood,
Nor scar that whiter skin of hers than snow,
And smooth as monumental alabaster.”
—William Shakespeare (1564–1616)
“Literature does not exist in a vacuum. Writers as such have a definite social function exactly proportional to their ability as writers. This is their main use.”
—Ezra Pound (1885–1972)
“The book borrower of real stature whom we envisage here proves himself to be an inveterate collector of books not so much by the fervor with which he guards his borrowed treasures and by the deaf ear which he turns to all reminders from the everyday world of legality as by his failure to read these books.”
—Walter Benjamin (1892–1940)
““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)