A function is called completely monotone if for every integer and every we have . (In particular, the derivatives must exist). For example, has this property, and so does . A completely monotone function is positive, decreasing, convex (all in the non-strict sense), and so are its derivatives of even orders. An innocent-looking calculus-type condition. Yet, it has some surprising consequences.
- Every completely monotone function is real-analytic on : that is, it is locally represented by its Taylor series
- Moreover, the Taylor series centered at has the radius of convergence : in other words, is the worst that can happen.
- Moreover, extends to a complex-analytic function on the right halfplane . Of course, this implies the items 1 and 3.
All of this follows at once from Bernstein’s theorem (1928): is completely monotone if and only if it is the Laplace transform of a positive measure: explicitly, if for some positive Borel measure on . The measure may be infinite, but the integral is required to converge for all . The case when is finite precisely corresponds to being bounded, and hence continuous on with . Some sources, including Wikipedia, deal only with the finite/bounded case. The complete treatment of Bernstein’s theorem, which also explains what kind of monotonicity it deals with, can be found in Barry Simon’s book Convexity: an analytic viewpoint which I heartily recommend.
This being a Calculus-oriented blog, let’s do an exercise: given , construct an infinitely differentiable function on which satisfies the condition for but not for . It's intuitively clear that such functions exist, but can we find a simple example?
A slight perturbation of the prototypical completely monotone function does the job: take
Let’s see: , …
Well, we could survive the computations in this form, but it’s better to use the identity and get everything done at once:
Now choose in the range .
A trigonometric aside. Compared to , the identity has very few fans. I remember seeing it in high school and shuddering from fear and disgust. What right had and to be there? Only much later did it dawn on me that can be taken as a new coordinate , and that the transformation , consists of scaling by and rotation by .