An Expression of (I-z )−1 by Means of Polynomials
Henry Frederick Baker · Proceedings of the London Mathematical Society · 1911
IT is well enough known (see, for instance, Borel, Series divergentes, 1901, p. 164) how to deduce a Mittag-Leffler star expansion of a monogenic function from a given explicit expansion of (1-z)~y of the same character.Some readers may be interested in having such an expansion actually set forth ; the method is, in part, merely a development of the original method of Runge {Ada Mathematica, vi, 1885, p. 287).Given in the plane of the complex variable z any region of finite dimensions containing no point infinitely near to any point of the real axis from z = 1 to z = -\-oo , it will be shewn how to form a series of polynomials converging uniformly in this region and representing (1-z)' 1 therein.Taking first a complex variable £, = f-H>;, enclose the points £ = 1, £ = 1 + c, wherein c is an arbitrary real positive quantity, by a closed curve consisting of (i) the straight lines >? = ± « , from £ = 1 to f = 1 + c, the quantity a being real and positive and arbitrary «. 1), (ii) a semicircle convex to the origin £ = 0 satisfying the equation (iii) a semicircle concave to the origin, of equation (£-1 -c) 2 +?/ 2 = a 2 .Keeping c and a fixed for the present, take a positive integer r so that c/ra is less than unity, = a-say; it is supposed that a is less than c, so that /• > 1 ; and take c o = l , c l = l + -, c 2 =l+-, ..., c r = l + c , so that the segment from £ = 1 to £ = l + c is divided into r equal parts.If )i x , >&2, ..., n r be positive integers, the rational function w