Nachbin's Theorem - Borel Transform

Borel Transform

Nachbin's theorem has immediate applications in Cauchy theorem-like situations, and for integral transforms. For example, the generalized Borel transform is given by

If f is of Ψ-type τ, then the exterior of the domain of convergence of, and all of its singular points, are contained within the disk

Furthermore, one has

where the contour of integration γ encircles the disk . This generalizes the usual Borel transform for exponential type, where . The integral form for the generalized Borel transform follows as well. Let be a function whose first derivative is bounded on the interval, so that

where . Then the integral form of the generalized Borel transform is

The ordinary Borel transform is regained by setting . Note that the integral form of the Borel transform is just the Laplace transform.