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Excerpt from Infinite Determinants in the Theory of Mathieu's and Hill's Equations The existence of even or odd solutions of Mathieu's equation with period n or 2n depends on the vanishing of certain infinite determinants. It is shown that these determinants can be expressed in terms of the values of the even or odd solutions at x n/z, where x is the independent variabhe. The corresponding result for Hill's determinant (which can be expressed in terms of the values of certain solutions at x a n) has long since been found. The method applied in the present report uses theorems about the order of growth of the solutions both with respect to x and the parameters. A rapra sentation of the solutions as a finite Fourier integral can also be derived by this method. We indicate how far the results can be extended to the case of more general equations of Hill's type. The periodic solutions of the inhomo goueoua Mathieu equations are constructed by using an inversion formula for. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

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