Cauchyn.
/ˈkəʊʃi/ Mathematics.
Used attributively and in the possessive to denote concepts introduced by Cauchy or arising from his work, as Cauchy distribution n. 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 n. (also 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 n. 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 n. spec. the theorem that the integral of any analytical function of a complex variable around a closed curve which encloses no singularities is zero.
ΘΚΠ
the world > relative properties > number > calculus > [noun] > integral calculus > integration or integrability > integral > theorem relating to
Cauchy's theorem1878
the world > relative properties > number > calculus > [noun] > integral calculus > integration or integrability > integral
fluent1706
integral1728
gamma function1834
surface integral1867
Riemann integral1894
Cauchy's integral1898
Lebesgue integral1905
Stieltjes integral1914
convolution1934
the world > relative properties > number > probability or statistics > [noun] > distribution
distribution1854
random distribution1882
frequency distribution1895
probability distribution1895
Poisson distribution1898
binomial distribution1911
Student's t-distribution1925
sampling distribution1928
probability density1931
Poisson1940
beta distribution1941
Cauchy distribution1948
geometric distribution1950
the world > relative properties > number > mathematical number or quantity > numerical arrangement > [noun] > set > sequence
subsequence1847
sequence1882
word1936
Lucas1953
Cauchy sequence1955
1878 Encycl. 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.
1889 Cent. Dict. at Formula Abel's, Cauchy's..formulæ, certain formulæ relating to definite integrals.
1893 J. Harkness & F. Morley Treat. Theory Functions v. 164 (heading) Cauchy's theorem.
1893 J. Harkness & F. Morley Treat. Theory Functions v. 167 Cauchy's theorem can be extended to functions which are holomorphic within a region bounded by more than one closed contour.
1898 Harkness & 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.
1910 Encycl. 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.
1932 E. C. Titchmarsh Theory of Functions ii. 81 This is Cauchy's integral formula.
1948 Ann. Math. Statistics 19 428 R0 is determined by integration over the upper tail of the Cauchy distribution.
1955 L. F. Boron tr. Natanson Theory of Functions of Real Variable I. vii. 171 The Bolzano–Cauchy property: every Cauchy sequence {xn} has a finite limit.
1962 Ann. Math. Statistics 33 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.
1982 W. 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.
1984 G. 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.
1984 G. B. Price Multivariable Anal. x. 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.
1987 Jones & Singerman Complex Functions 318 The basic theorem of complex integration is Cauchy's theorem.
This entry has not yet been fully updated (first published 1997; most recently modified version published online June 2019).
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