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Wikipedia / Cauchy's integral formula

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, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.

Theorem

be an open subset of the complex plane is completely contained in be a holomorphic function be the circle, oriented counterclockwise, forming the boundary of in the interior of

The proof of this statement uses the Cauchy integral theorem to be complex differentiable can be expanded as a power series it follows that holomorphic functions are analytic

This formula is sometimes referred to as Cauchy's differentiation formula.

can be replaced by any closed rectifiable curve that has winding number one about be holomorphic in the open region enclosed by the path and continuous on its closure.

Note that not every continuous function on the boundary can be used to produce a function inside the boundary that fits the given boundary function. For instance, if we put the function , defined for , into the Cauchy integral formula, we get zero for all points inside the circle. In fact, giving just the real part on the boundary of a holomorphic function is enough to determine the function up to an imaginary constant – there is only one imaginary part on the boundary that corresponds to the given real part, up to addition of a constant. We can use a combination of a Möbius transformation and the Stieltjes inversion formula has real part . On the unit circle this can be written term makes no contribution, and we find the function . This has the correct real part on the boundary, and also gives us the corresponding imaginary part, but off by a constant, namely .

Proof sketch

By using the Cauchy integral theorem (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around is arbitrarily close to centered at . This can be calculated directly via a parametrization (integration by substitution is the radius of the circle.

Example

(the circle of radius 2).

around the contour , we need to know the singularities of and .

and . The moduli of these points are less than 2 and thus lie inside the contour. This integral can be split into two smaller integrals by Cauchy–Goursat theorem where the contour is a small circle around each pole around .

Now, each of these smaller integrals can be evaluated by the Cauchy integral formula, but they first must be rewritten to apply the theorem. For the integral around as . This is analytic

An elementary trick using partial fraction decomposition

Consequences

The integral formula has broad applications. First, it implies that a function that is holomorphic in an open set is in fact infinitely differentiable there. Furthermore, it is an analytic function, meaning that it can be represented as a power series. The proof of this uses the dominated convergence theorem and the geometric series

The formula is also used to prove the residue theorem, which is a result for meromorphic functions, and a related result, the argument principle. It is known from Morera's theorem that the uniform limit of holomorphic functions is holomorphic. This can also be deduced from Cauchy's integral formula: indeed the formula also holds in the limit and the integrand, and hence the integral, can be expanded as a power series. In addition the Cauchy formulas for the higher order derivatives show that all these derivatives also converge uniformly.

The analog of the Cauchy integral formula in real analysis is the Poisson integral formula for harmonic functions; many of the results for holomorphic functions carry over to this setting. No such results, however, are valid for more general classes of differentiable or real analytic functions. For instance, the existence of the first derivative of a real function need not imply the existence of higher order derivatives, nor in particular the analyticity of the function. Likewise, the uniform limit of a sequence of (real) differentiable functions may fail to be differentiable, or may be differentiable but with a derivative that is not the limit of the derivatives of the members of the sequence.

satisfy Cauchy's estimate

From Cauchy's estimate, one can easily deduce that every bounded entire function must be constant (which is Liouville's theorem).

The formula can also be used to derive Gauss's mean-value theorem, which states

is . This can be calculated directly via a parametrization of the circle.

Generalizations

Smooth functions

A version of Cauchy's integral formula is the Cauchy–Pompeiu formula, and holds for smooth functions as well, as it is based on Stokes' theorem is a complex-valued function on the closure of . Then

One may use this representation formula to solve the inhomogeneous Cauchy–Riemann equations in is a function in of the equation is a holomorphic function outside the support of . Moreover, if in an open set (where

The first conclusion is, succinctly, that the convolution of a compactly supported measure with the Cauchy kernel is a holomorphic function off the support of denotes the principal value. The second conclusion asserts that the Cauchy kernel is a fundamental solution and is a restatement of the fact that, considered as a distribution is a fundamental solution of the Cauchy–Riemann operator .

from this result and the formula for the distributional derivative of the characteristic function of where the distribution on the right hand side denotes contour integration along .

Now we can deduce the generalized Cauchy integral formula:

Several variables

In several complex variables, the Cauchy integral formula can be generalized to polydiscs. be the polydisc given as the Cartesian product open discs

continuous on the closure of where .

In real algebras

The Cauchy integral formula is generalizable to real vector spaces of two or more dimensions. The insight into this property comes from geometric algebra, where objects beyond scalars and vectors (such as planar bivectors and volumetric trivectors) are considered, and a proper generalization of Stokes' theorem.

-vector field and generally has in its derivative a scalar part, the divergence (), and a bivector part, the curl (). This particular derivative operator has a Green's function-ball in the space (that is, , the circumference of a circle with radius , and , the surface area of a sphere with radius

It is this useful property that can be used, in conjunction with the generalized Stokes theorem

When is called a monogenic function-vector, the pseudoscalar

Thus, as in the two-dimensional (complex analysis) case, the value of an analytic (monogenic) function at a point can be found by an integral over the surface surrounding the point, and this is valid not only for scalar functions but vector and general multivector functions as well.

See also

Notes

References