'Is Regular.2 We Intend
VOL. 41, 1955 MATHEMATICS: P. SAMUEL 647 * This work was carried out at the University of Wisconsin Naval Research Laboratory under Contract No. AT(11-1)-298 with the United States Atomic Energy Commission. 'Some authors, including MacRoberts, Magnus, and Oberhettinger, define Ferrers' associated Legendre function of the first kind as (1)' times the definition given in eqt (2). Our definition, however, agrees with Jahnke and Emde and with Whittaker and Watspn. 2 H. Margenau and G. M. Murphy, Mathematics of Physics and Chenmistry (New York: D. Van Nostrand Co., 1943), p. 103, eq. (3-45). 3A. Erdelyi, "Higher Transcendental Functions" (Bateman Manuscript Project) (McGraw- Hill, 1953) 1, 172, eq. (27). See also T. M. MacRoberts, Quart. J. Math, (Oxford ser.), 11, 95 and 96, 1940; T. M. MacRoberts, Spherical Harmonics (New York: Dover Publications, Inc., 1947), p. 367, ex. 86(ii), together with p. 143, ex. 10, and p. 125, eq. (14). SIMPLE SUBVARIETIES OF ANALYTIC VARIETIES* BY PIERRE SAMUEL HARVARD UNIVERSITY AND UNIVERSITY OF CLERMONT-FERRAND Communicated by Oscar Zariski, June 2, 1955 1. Inttoduction.-Let k be a field. We make the simplifying assumption that its characteristic p is not 0. To every prime ideal p in the power series ring A = kI[[X, . ., X.]] we associate an object V(p) called an analytic variety over U.' The dimension of the local ring A/p is called the dimension of V(p).2 If p c p17 we say that V(p') is a subvariety of V(p). A subvariety V(p') of an analytic variety V(p) is called simple if its local ring (A/A),yi,,= is' regular.2 We intend to show that Zariski's generalized Jacobian criterion for simple subvarieties of algebraic varieties may be extended to analytic varieties.
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