翻訳と辞書 Words near each other ・ Laurent Romejko ・ Laurent Ronde ・ Laurent Roussey ・ Laurent Roux ・ Laurent Ruquier ・ Laurent Rédon ・ Laurent Sagart ・ Laurent Salgues ・ Laurent Saloff-Coste ・ Laurent Sauvagnac ・ Laurent Schwartz ・ Laurent Schwartz (oncologist) ・ Laurent Sciarra ・ Laurent Sedego ・ Laurent Sempéré ・ Laurent series ・ Laurent Simutoga ・ Laurent Sky ・ Laurent Stocker ・ Laurent Tailhade ・ Laurent Terzieff ・ Laurent Tessier ・ Laurent Thévenot ・ Laurent Tirard ・ Laurent Tobel ・ Laurent Touil-Tartour ・ Laurent Touyeras ・ Laurent Travini ・ Laurent Tuel ・ Laurent Vallon Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Laurent series ： ウィキペディア英語版 Laurent series In mathematics, the Laurent series of a complex function ''f''(''z'') is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first but his paper, written in 1841, was not published until much later, after Weierstrass' death.〔.〕 The Laurent series for a complex function ''f''(''z'') about a point ''c'' is given by: :$f(z)=\backslash sum\_^\backslash infty\; a\_n(z-c)^n$ where the ''an'' are constants, defined by a line integral which is a generalization of Cauchy's integral formula: :$a\_n=\backslash frac\; \backslash oint\_\backslash gamma\; \backslash frac\{(z-c)^\{n+1\}\}.\backslash ,$ The path of integration $\backslash gamma$ is counterclockwise around a closed, rectifiable path containing no self-intersections, enclosing ''c'' and lying in an annulus ''A'' in which $f(z)$ is holomorphic (analytic). The expansion for $f(z)$ will then be valid anywhere inside the annulus. The annulus is shown in red in the diagram on the right, along with an example of a suitable path of integration labeled $\backslash gamma$. If we take $\backslash gamma$ to be a circle $|z-c|\; =\; \backslash varrho$, where , this just amounts to computing the complex Fourier coefficients of the restriction of $f$ to $\backslash gamma$. The fact that these integrals are unchanged by a deformation of the contour $\backslash gamma$ is an immediate consequence of Green's theorem. In practice, the above integral formula may not offer the most practical method for computing the coefficients $a\_n$ for a given function $f(z)$; instead, one often pieces together the Laurent series by combining known Taylor expansions. Because the Laurent expansion of a function is unique whenever it exists, any expression of this form that actually equals the given function $f(z)$ in some annulus must actually be the Laurent expansion of $f(z)$. == Convergent Laurent series == Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities. Image:Expinvsqlau SVG.svgan style="color:#b3b300;">4, 5, 6, 7 and 50. As ''N'' → ∞, the approximation becomes exact for all (complex) numbers ''x'' except at the singularity ''x'' = 0. More generally, Laurent series can be used to express holomorphic functions defined on an annulus, much as power series are used to express holomorphic functions defined on a disc. Suppose :$\backslash sum\_^\backslash infty\; a\_n\; (\; z\; -\; c\; )^n$ is a given Laurent series with complex coefficients ''a''''n'' and a complex center ''c''. Then there exists a unique inner radius r and outer radius ''R'' such that: * The Laurent series converges on the open annulus ''A'' ≡ . To say that the Laurent series converges, we mean that both the positive degree power series and the negative degree power series converge. Furthermore, this convergence will be uniform on compact sets. Finally, the convergent series defines a holomorphic function ''ƒ''(''z'') on the open annulus. * Outside the annulus, the Laurent series diverges. That is, at each point of the exterior of ''A'', the positive degree power series or the negative degree power series diverges. * On the boundary of the annulus, one cannot make a general statement, except to say that there is at least one point on the inner boundary and one point on the outer boundary such that ''ƒ''(''z'') cannot be holomorphically continued to those points. It is possible that ''r'' may be zero or ''R'' may be infinite; at the other extreme, it's not necessarily true that ''r'' is less than ''R''. These radii can be computed as follows: :$\backslash begin$ r &= \limsup_ |a_|^\frac \\ &= \limsup_ |a_n|^\frac. \end We take ''R'' to be infinite when this latter lim sup is zero. Conversely, if we start with an annulus of the form ''A'' ≡ and a holomorphic function ''ƒ''(''z'') defined on ''A'', then there always exists a unique Laurent series with center ''c'' which converges (at least) on ''A'' and represents the function ''ƒ''(''z''). As an example, let :$f(z)\; =\; .$ This function has singularities at ''z'' = 1 and ''z'' = 2''i'', where the denominator of the expression is zero and the expression is therefore undefined. A Taylor series about ''z'' = 0 (which yields a power series) will only converge in a disc of radius 1, since it "hits" the singularity at 1. However, there are three possible Laurent expansions about 0, depending on the region ''z'' is in: * One is defined on the disc where |''z''| < 1; it is the same as the Taylor series, ::$f(z)\; =\; \backslash frac\; \backslash sum\_^\backslash infty\; \backslash left(\backslash frac\; \backslash left(\backslash sum\_^\backslash infty\; \backslash frac\; +\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash sum\_^\backslash infty\; \backslash frac.$ *:(The terms above can be derived through polynomial long division or using the sum of a geometric series trick again, this time using $\backslash frac$ and $\backslash frac$ as the common ratios.) The case ''r'' = 0; i.e., a holomorphic function ''ƒ''(''z'') which may be undefined at a single point ''c'', is especially important. The coefficient ''a''−1 of the Laurent expansion of such a function is called the residue of ''ƒ''(''z'') at the singularity ''c''; it plays a prominent role in the residue theorem. For an example of this, consider :$f(z)\; =\; +\; e^\backslash frac.$ This function is holomorphic everywhere except at ''z'' = 0. To determine the Laurent expansion about ''c'' = 0, we use our knowledge of the Taylor series of the exponential function: :$f(z)\; =\; \backslash cdots\; +\; \backslash left(\; \backslash right)\; z^\; +\; \backslash left(\; \backslash right)\; z^\; +\; 2z^\; +\; 2\; +\; \backslash left(\; \backslash right)\; z\; +\; \backslash left(\; \backslash right)\; z^2\; +\; \backslash left(\; \backslash right)\; z^3\; +\; \backslash cdots$ and we find that the residue is 2. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Laurent series」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.