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Laurent, n. Math.|lɒˈrɑ̃|
[The name of P. A. Laurent (1813–54), French mathematician.]
Used attrib. and in the possessive in Laurent('s) expansion, series, the expression of a function of a complex variable z as a power series in (z - a) , where a is a fixed point; Laurent's theorem, the theorem that a single-valued monogenic analytic function may be expressed as a Laurent series in (z - a) at all points of any annular region centred on a and lying entirely within the domain of existence of the function.
1893A. R. Forsyth Theory Functions Complex Variable iii. 47 (heading) Laurent's expansion of a function. Ibid., Laurent's theorem is as follows:—A function, which is holomorphic in a part of the plane bounded by two concentric circles with centre a and finite radii, can be expanded in the form of a double series of integral powers, positive and negative, of z – a, the series converging uniformly and unconditionally in the part of the plane between the circles. 1898Harkness & Morley Introd. Theory Analytic Functions x. 125 We shall have to consider series in both ascending and descending powers, such as a0 + a1(x - c) + a2(x - c)2 +...+ a-1(x - c)-1 + a -2(x - c)-2+.... These need no new notation, as they can be expressed by P(x - c) + P(1/(x - c)) . The two constitute a Laurent series. 1932E. C. Titchmarsh Theory of Functions xiii. 401 There is a close formal connexion between a Fourier series and a Laurent series. 1968P. A. P. Moran Introd. Probability Theory x. 479 We expand this in a Laurent series convergent in a ring 0 〈 α 〈 {vb}z{vb} 〈 β. Ibid., The Laurent expansion of (10.70) is therefore g0(z, s) = [etc.] . 1974P. Henrici Appl. & Computational Complex Analysis I. iv. 211 Laurent's theorem contains the Taylor expansion as a special case, and a remarkable conclusion may be drawn from it. 1990Proc. London Math. Soc. LXI. iii. 546 We shall get the finiteness of the Laurent expansion in λ of the based extended solution ϕλ for a pluriharmonic map from a compact complex manifold. |