# Cauchy integral theorem

In mathematics, the

The theorem is usually formulated for closed paths as follows: let

As was shown by Goursat, Cauchy's integral theorem can be proven assuming only that the complex derivative

The condition that

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

is non-zero; the Cauchy integral theorem does not apply here since

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

The Cauchy integral theorem is valid in a slightly stronger form than given above. Suppose

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

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.

We can write

Now using Green's theorem, we can write:

and

Since

and

Substituting in

and

or, finally, that

Augustin Louis Cauchy

Born 21 July 1789

Dijon, France

Died 23 May 1857 (aged 69)

A common example is the absolute value function, given by

**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*:*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,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; considerwhich 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*:*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*, thenThe 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*, thenThe 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**R**^{n}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**R**^{n}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*, thenWe can write

*f*(*z*) as*f*(*x*+*iy*) =*u*(*x*,*y*) +*iv*(*x*,*y*). Then we haveNow 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 getand

or, finally, that

## See also

## External links

- Eric W. Weisstein,
*Cauchy Integral Theorem*at MathWorld. - Cauchy-Goursat Theorem Module by John H. Mathews

**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".

**.....**Click the link for more information.

**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

Dijon, France

Died 23 May 1857 (aged 69)

**.....**Click the link for more information.

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**.**.....**Click the link for more information.**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.

**.....**Click the link for more information.

In mathematics, a

where

**complex number**is a number of the formwhere

*a*and*b*are real numbers, and*i*is the imaginary unit, with the property*i*² = −1.**.....**Click the link for more information. 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*.**.....**Click the link for more information. 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.**.....**Click the link for more information. 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.**.....**Click the link for more information.**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.**.....**Click the link for more information. 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.**.....**Click the link for more information.**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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For the

In mathematics, the **fundamental group of a factor**see von Neumann algebra.**fundamental group**is one of the basic concepts of algebraic topology.

**.....**Click the link for more information.

The

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite

**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

**.....**Click the link for more information. 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.**.....**Click the link for more information. 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*.**.....**Click the link for more information.**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

**.....**Click the link for more information.

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*.**.....**Click the link for more information.**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*.

**.....**Click the link for more information.

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.**.....**Click the link for more information. 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.**.....**Click the link for more information. 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.**.....**Click the link for more information.**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**.....**Click the link for more information.**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*.

**.....**Click the link for more information.

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.**.....**Click the link for more information. 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*.**.....**Click the link for more information. 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.**.....**Click the link for more information. 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.**.....**Click the link for more information.This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.