Cauchy-Heine Transform

Abstract

Let Ψ(w) be a continuous function for w on the straight line segment from 0 to a point a≠0. Then the function

$$ f\left( z \right) = \frac{1} {{2\pi i}}\int_0^a {\frac{{\psi \left( w \right)}} {{w - z}}} dw $$

obviously is holomorphic for z in the complex plane with a cut from 0 to a, but f in general will be singular at points on this cut. If Ψ is holomorphic, at least at points strictly between 0 and a, then one can use Cauchy’s integral formula to see that f is holomorphic at these points, too. At the endpoints, however, f will be singular, even if Ψ is analytic there; for this, see the exercises at the end of the first section. For our purposes, it will be important to assume that Ψ, for w → 0, decreases faster than arbitrary powers of w, since then we shall see that f will have an asymptotic power series expansion at the origin. We shall even show that this expansion is of Gevrey order s, provided that $\psi \left( w \right) \cong _s \hat 0$ . Hence integrals of the above type provide an excellent tool for producing examples of functions with asymptotic expansions, or even of series that are κ-summable in certain directions. Much more can be done, however: For arbitrary functions, analytic in a sectorial region and having an asymptotic expansion at the origin, we shall obtain a representation that is the analogue to Cauchy’s formula for functions analytic at the origin. As a special case, we shall obtain a very useful characterization of such functions f that are the sums of ]κ — summable series in some direction d. In other words, we shall characterize the image of the operator S _k,d , for κ > 0 and d ε ℝ.