Laurentn.

/lɒˈrɒ̃/

Etymology: < the name of P. A. Laurent (1813–54), French mathematician.

Mathematics.

Used attributively and in the possessive in Laurent's expansion n. (also Laurent expansion) , 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 series n. (also Laurent series) = Laurent's expansion n. Laurent's theorem n. 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.

ΘΚΠ

the world > relative properties > number > mathematics > [noun] > mathematical enquiry > proposition > theorem > specific theorem > relating to functions

Taylor's theorem1816

Maclaurin's theorem1820

Leibniz theorem1852

Green's theorem1857

Laurent's theorem1893

factor theorem1894

factor law1901

the world > relative properties > number > mathematical number or quantity > numerical arrangement > [noun] > set > sequence > series > infinite

secundan1685

infinite series1706

Taylor('s) series1816

Maclaurin's series1881

power series1884

Fibonacci('s) series1891

Laurent's expansion1893

Fibonacci('s) numbers1914

majorant1925

tetrahedral numbers1939

Fibonacci('s) sequence1964

binomial series1966

1893 A. R. Forsyth Theory Functions Complex Variable iii. 47 (heading) Laurent's expansion of a function.

1893 A. R. Forsyth Theory Functions Complex Variable iii. 47 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.

1898 Harkness & 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.

1932 E. C. Titchmarsh Theory of Functions xiii. 401 There is a close formal connexion between a Fourier series and a Laurent series.

1968 P. A. P. Moran Introd. Probability Theory x. 479 We expand this in a Laurent series convergent in a ring 0 〈 α 〈 |z| 〈 β.

1968 P. A. P. Moran Introd. Probability Theory x. 479 The Laurent expansion of (10.70) is therefore g₀(z, s) =[etc.].

1974 P. 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.

1990 Proc. London Math. Soc. 61 546 We shall get the finiteness of the Laurent expansion in λ of the based extended solution Φ_λ for a pluriharmonic map from a compact complex manifold.

This entry has not yet been fully updated (first published 1997; most recently modified version published online June 2019).

单词	laurent
释义	Laurentn. /lɒˈrɒ̃/ Etymology: < the name of P. A. Laurent (1813–54), French mathematician. Mathematics. Used attributively and in the possessive in Laurent's expansion n. (also Laurent expansion) , 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 series n. (also Laurent series) = Laurent's expansion n. Laurent's theorem n. 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. ΘΚΠ the world > relative properties > number > mathematics > [noun] > mathematical enquiry > proposition > theorem > specific theorem > relating to functions Taylor's theorem1816 Maclaurin's theorem1820 Leibniz theorem1852 Green's theorem1857 Laurent's theorem1893 factor theorem1894 factor law1901 the world > relative properties > number > mathematical number or quantity > numerical arrangement > [noun] > set > sequence > series > infinite secundan1685 infinite series1706 Taylor('s) series1816 Maclaurin's series1881 power series1884 Fibonacci('s) series1891 Laurent's expansion1893 Fibonacci('s) numbers1914 majorant1925 tetrahedral numbers1939 Fibonacci('s) sequence1964 binomial series1966 1893 A. R. Forsyth Theory Functions Complex Variable iii. 47 (heading) Laurent's expansion of a function. 1893 A. R. Forsyth Theory Functions Complex Variable iii. 47 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. 1898 Harkness & 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. 1932 E. C. Titchmarsh Theory of Functions xiii. 401 There is a close formal connexion between a Fourier series and a Laurent series. 1968 P. A. P. Moran Introd. Probability Theory x. 479 We expand this in a Laurent series convergent in a ring 0 〈 α 〈 \|z\| 〈 β. 1968 P. A. P. Moran Introd. Probability Theory x. 479 The Laurent expansion of (10.70) is therefore g₀(z, s) =[etc.]. 1974 P. 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. 1990 Proc. London Math. Soc. 61 546 We shall get the finiteness of the Laurent expansion in λ of the based extended solution Φ_λ for a pluriharmonic map from a compact complex manifold. This entry has not yet been fully updated (first published 1997; most recently modified version published online June 2019). < n.1893
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