Harry Hochstadt epub Addition Theorems for the Functions of the Paraboloid of Revolution (Classic Reprint)

Introduction The wave equation U+ k- 0admits solutions of the foim if the coordinate system is such that separation of variables is possible. 4, 7] and 0are the three independent variables, and xand [i represent arbitrary complex parameters. In general Uwill not be reguJ.ar and one-valued over the whole space, but will be so for special values of X- and l. Let Kt Vand 0be functions of 4, 11, and resvilting from a translation or rotation of the coordinate system; then a relation which expresses U(C ,V) ,)d )as a summation of teiTO sof the form U(C ri,)!)) is called an addition theorem. Addition theorems for cylindrical and spherical coordinate systems are well known. These are the addition theorems forB essel and Hankel functions, Legendre polynomials, spherical harmonics, Mathieu functions and spheroidal wave functions (see Meixner andS chafke -and Erdelyi -- ). It is proposed to derive such addition theorems for those functions of the paraboloid of revolution which are regular and one-valued in the whole space. As will be seen subsequently, these restrictions are not always necessary. That such theorems should exist can be inferred from the invariance of Uunder rotations and translations of the space, and from the fact that the family of solutions that are everywhere regular and one-valued will be mapped onto itself by motions of space It is possible to derive several of these theorems by using known addition theorems. For example, it is possible to derive linear relations between the functions of the paraboloid of revolution and spherical harmonics.(Typographical errors above are due to OCR software and don't occur in the book.)About the Publisher Forgotten Books is a publisher of historical writings, such as: Philosophy, Classics, Science, Religion, History, Folklore and Mythology.Forgotten Books' Classic Reprint Series utilizes the latest technology to regenerate facsimiles