In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin Cauchy, is an important statement about path integrals for holomorphic functions in the complex plane. Essentially, it says that if two different paths connect the same two points, and a function is holomorphic everywhere "in between" the two paths, then the two path integrals of the function will be the same. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

The theorem is usually formulated for closed paths as follows: let U be an open subset of C which is simply connected, let f : U -> C be a holomorphic function, and let γ be a rectifiable path in U whose start point is equal to its end point. Then, In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...

As was shown by Goursat, Cauchy's integral theorem can be proven assuming only that the complex derivative f '(z) exists everywhere in U. This is significant, because one can then prove Cauchy's integral formula for these functions, and from that one can deduce that these functions are in fact infinitely often continuously differentiable. Edouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his Cours danalyse mathématique, which appeared in the first decade of the twentieth century. ... In mathematics, Cauchys integral formula, named after Augustin Cauchy, is a central statement in complex analysis. ...

The condition that U be simply connected means that U have no "holes"; for instance, every open disk U = { z : |z - z₀| < r } qualifies. The condition is crucial; for example, if

where exp() is the exponential function traces out the unit circle, then the path integral The exponential function is one of the most important functions in mathematics. ...

is non-zero; the Cauchy integral theorem does not apply here since f(z) = 1/z is not defined (and certainly not holomorphic) at z = 0.

One important consequence of the theorem is that path integrals of holomorphic functions on simply connected domains can be computed in a manner familiar from the fundamental theorem of real calculus: let U be a simply connected open subset of C, let f : U -> C be a holomorphic function, and let γ be a piecewise continuously differentiable path in U with start point a and end point b. If F is a complex antiderivative of f, then The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...

The Cauchy integral theorem is valid in a slightly stronger form than given above. Suppose U is an open simply connected subset of C whose boundary is the image of the rectifiable path γ. If f is a function which is holomorphic on U and continuous on the closure of U, then For a different notion of boundary related to manifolds, see that article. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...

The Cauchy integral theorem is considerably generalized by the Cauchy integral formula and the residue theorem. Cauchys integral formula is a central statement in complex analysis. ... The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ...

See also Morera's theorem. In complex analysis, Moreras theorem states that if the integral of a continuous complex-valued function f of a complex variable along every simple closed curve within an open set is zero, that is, if for C any simple closed curve, then f is differentiable at every point in...