Differential resolvents of minimal order and weight

We will determine the number of powers ofαthat appear with nonzero coefficient in anα-power linear differential resolvent of smallest possible order of a univariate polynomialP(t)whose coefficients lie in an ordinary differential field and whose distinct roots are differentially independent over constants. We will then give an upper bound on the weight of anα-resolvent of smallest possible weight. We will then compute the indicial equation, apparent singularities, and Wronskian of the Cockleα-resolvent of a trinomial and finish with a related determinantal formula.

Holomorphic solutions about an irregular singular point of an ordinary linear differential equation

It is shown that that an ordinary linear differential equation may possess a holomorphic solution in a neighbourhood of an irregular singular point even though the usual linearly independent solutions corresponding to the two roots of the indicial equation both have zero radius of convergence.

Shear and rotation moiré patterns obtained by spatial filtering of diffraction patterns

Two superposed gratings can produce many other moiré patterns in addition to the pattern commonly observed. They correspond to different forms of the indicial equation employed in parametric descriptions of moiré phenomena. An analysis of the Fraunhofer diffraction pattern of two superposed gratings by the methods of Fourier optics shows that the different moiré patterns can be separately observed by spatial-filtering techniques. Examples of additive and subtractive moiré patterns obtained individually over the whole field of two superposed gratings are presented. This method is combined with a previously developed method for obtaining partial derivatives from deformed gratings as a direct moiré pattern (rather than as moiré of moiré) to determine experimentally the whole field of cartesian shears and rigid rotations. The method is applied to a circular ring subjected to diametral compression.

Separation of Additive and Subtractive Moire´ Patterns

Using a spatial filtering technique, moire´ patterns can be observed that correspond to different forms of the indicial equation describing the parametric properties of gratings and moire´. In particular both the additive and subtractive moire´s are obtained individually over the field. This method, combined with a previously developed method for obtaining spatial partial derivatives from deformed gratings, permits the experimental determination of the whole field of Cartesian shears and rigid rotations.

Univalent Solutions of W″ + pW = 0

Consider the differential equation1.1where1.2is regular in |z| < R.The indicial equation associated with (1.1) is of the formWe shall denote the two roots of this equation by α and β, where . Corresponding to the root α there exists a unique solution of (1.1) of the following form1.3In those cases for which there exists a unique second solution of (1.1) of the form1.4