Cauchy integral theorem

In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin Louis Cauchy, is an important statement about line 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.

The theorem is usually formulated for closed paths as follows: let U be an open subset of C which is simply connected, let f : UC be a holomorphic function, and let γ be a rectifiable path in U whose start point is equal to its end point. Then,



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.

The condition that U be simply connected means that U has no "holes" or, in homotopy terms, that the fundamental group of U is trivial; for instance, every open disk qualifies. The condition is crucial; consider



which traces out the unit circle, and then the path integral



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 : UC 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 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



The Cauchy integral theorem leads to the Cauchy integral formula and the residue theorem.

A real-plane generalization can be done using the Stokes theorem:



applied to an open set M in Rn whose boundary is a smooth manifold. If the form is exact so and its radial part in polar coordinates diverges as extracting a ball B from the manifold M and integrating on M\B, then making the radius of the ball tend to 0 we get a similar result to Cauchy's integral theorem on Rn in the form:



On page 513 of the book "vector calculus" by Marsden-Tromba, Gauss applies this formulation to the 3-D gravitational field on an arbitrary surface to give an example of his divergence theorem.

Proof

We shall prove the statement that if U is an open simply connected subset of C whose boundary is the image of the rectifiable path γ and f is a function which is holomorphic on U and continuous on the closure of U, then



We can write f(z) as f(x+iy) = u(x,y) + iv(x,y). Then we have



Now using Green's theorem, we can write:



and



Since f is holomorphic in U, the partial derivatives of u and v are related with the Cauchy-Riemann equations :



and



Substituting in (1) and (2) we get



and



or, finally, that

See also

External links

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics.
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Augustin Louis Cauchy

Augustin Louis Cauchy
Born 21 July 1789(1789--)
Dijon, France
Died 23 May 1857 (aged 69)
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In mathematics, a line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use. In the case of a closed curve it is also called a contour integral.
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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.
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In mathematics, a complex number is a number of the form


where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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In topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U.
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In topology, a geometrical object or space is called simply connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed into every other.
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In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle.
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Edouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his Cours d'analyse mathématique, which appeared in the first decade of the twentieth century.
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In mathematics, Cauchy's integral formula, named after Augustin Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk.
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In topology, a geometrical object or space is called simply connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed into every other.
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homotopic (Greek homos = identical and topos = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
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In mathematics, the fundamental group is one of the basic concepts of algebraic topology.
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The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite
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In topology, a geometrical object or space is called simply connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed into every other.
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In topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U.
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piecewise-defined function f(x) of a real variable x is a function whose definition is given differently on disjoint subsets of its domain.

A common example is the absolute value function, given by

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In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function g is a function whose complex derivative is g.
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boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of S.
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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.
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In mathematics, Cauchy's integral formula, named after Augustin Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk.
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The residue theorem in complex analysis is a powerful tool to evaluate line integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula.
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Stokes' theorem (or Stokes's theorem) in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus.
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In vector calculus, the divergence theorem, also known as Gauss' theorem, Ostrogradsky's theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the
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boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of S.
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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.
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In physics and mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C.
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In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic.
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In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic.
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