| 释义 |
Cauchy, n. Math.|ˈkəʊʃɪ|
[The name of Augustin-Louis Cauchy (1789–1857), French mathematician.]
Used attrib. and in the possessive to denote concepts introduced by Cauchy or arising from his work, as Cauchy distribution, a probability distribution whose probability density is of the form a/[1+b(x-λ)2] , where λ is the median of the distribution and a and b are constants; Cauchy's integral (formula), a formula expressing the value of a function f(z) at a point a in terms of an integral around a closed curve enclosing it, which may be written ∫[f(z)/(z-a)]dz = 2πif(a)k , where k has the value 1 if a is inside the curve, 0 if outside, and ½ if a lies on it; Cauchy sequence, any sequence of numbers an such that for any positive number ε, a value of n can be chosen so that any two members of the sequence after an differ by a quantity whose magnitude is less than ε; Cauchy's theorem, spec. the theorem that the integral of any analytical function of a complex variable around a closed curve which encloses no singularities is zero.
1878Encycl. Brit. VIII. 503/1 Cauchy's theorem contains really the proof of the fundamental theorem that a numerical equation of the nth order..has precisely n roots. 1889Cent. Dict. s.v. Formula, Abel's, Cauchy's..formulæ, certain formulæ relating to definite integrals. 1893Harkness & Morley Treat. Theory of Functions v. 164 (heading) Cauchy's theorem. Ibid. 167 Cauchy's theorem can be extended to functions which are holomorphic within a region bounded by more than one closed contour. 1898Harkness & Morley Introd. Theory Analytic Functions xvi. 222 The integral.. (1/2πi)∫fxdx/(Ax-c) , taken over a circuit A in Γ, may be called Cauchy's integral, for it plays an essential part in the development of the theory of functions along Cauchy's lines. 1910Encycl. Brit. XI. 314/1 The theory of the integration of a monogenic function, and Cauchy's theorem, that ∫f(z)dz = 0 over a closed path, are at once deducible from the corresponding results applied to a single power series for the interior of its circle of convergence. 1932E. C. Titchmarsh Theory of Functions ii. 81 This is Cauchy's integral formula. 1948Ann. Math. Statistics XIX. 428 R0 is determined by integration over the upper tail of the Cauchy distribution. 1955L. F. Boron tr. Natanson's Theory of Functions of Real Variable I. vii. 171 The Bolzano–Cauchy property: every Cauchy sequence {ob}xn{cb} has a finite limit. 1962Ann. Math. Statistics XXXIII. 1258 Given a symmetric bivariate Cauchy distribution and knowledge of its marginal distributions in several directions, one would like to know what possibilities are available for the marginal distributions in a few other directions. 1982W. S. Hatcher Logical Found. Math. v. 181 Completeness means that all Cauchy sequences converge or (equivalently for totally ordered fields) that every bounded nonempty set of elements of the field has a least upper bound. 1984G. B. Price Multivariable Anal. x. 521 This section uses Theorem 67.2, Goursat's form of Cauchy's integral theorem, to prove Cauchy's integral formula. Ibid. 526 This formula is easy to remember because it can be derived from Cauchy's integral formula..by differentiating under the integral sign n times with respect to z. 1987Jones & Singerman Complex Functions 318 The basic theorem of complex integration is Cauchy's theorem. |