On multiform solutions of linear differential equations having elliptic function coefficients
Transactions of the American Mathematical Society
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use W. L. MISER : ON MULTIFORM SOLUTIONS [April sary and sufficient conditions are known that all the solutions shall be uniform in the vicinity of such a singular point. Since Picard many writers have treated the class of linear differential equations having elliptic function coefficients, some of them making studies of the solutions of certain particular equations. A list of their treatises
... of their treatises and memoirs is given in the bibliography at the end of this paper. Among the most important of the memoirs are those by Floquet,6 Stenberg,12 Plemelj,18 and Mercer.20 There are two directions in which Mercer in his paper makes his problem less restricted than that considered by all earlier writers. In the first place he does not limit himself to the case in which the coefficients are mere elliptic functions, but adopts the wider condition that the coefficients shall have for their singularities a reducible set of points. In the second place he assumes that the solutions are all uniform when considered as localized in a doublyperiodic region $ which excludes a region 0 and its congruent regions obtained by shrinking the sides of a pseudo-parallelogram* of periods so as not to pass over any singularities of the coefficients. The present investigation has to do only with a system of n linear homogeneous differential equations of the first order whose coefficients are elliptic functions having the common periods 2co and 2co', and having only simple poles. The hypothesis that the coefficients are elliptic functions, rather than uniform doubly-periodic functions, which if non-elliptic have essential singularities in the finite portion of the plane, is made in order that, in each common parallelogram of periods, all the singular points of the solutions shall be isolated, and, therefore, finite in number. The elliptic functions are restricted to have only simple poles in order that, in the whole finite portion of the plane of the independent variable, all the singular points of the solutions shall be singular points of determination, and in order that use can, therefore, be made of the general theory of Fuchs respecting the nature of solutions of linear differential equations in the vicinity of such singular points.