Mathematical analysis → Complex analysis |

Complex analysis |
---|

Complex numbers |

Complex functions |

Basic Theory |

Geometric function theory |

People |

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.

Let *U* be an open subset of the complex plane **C**, and suppose the closed disk *D* defined as

is completely contained in *U*. Let *f* : *U* → **C** be a holomorphic function, and let *γ* be the circle, oriented counterclockwise, forming the boundary of *D*. Then for every *a* in the interior of *D*,

The proof of this statement uses the Cauchy integral theorem and like that theorem, it only requires *f* to be complex differentiable. Since can be expanded as a power series in the variable :

— it follows that holomorphic functions are analytic, i.e. they can be expanded as convergent power series. In particular *f* is actually infinitely differentiable, with

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

The theorem stated above can be generalized. The circle *γ* can be replaced by any closed rectifiable curve in *U* which has winding number one about *a*. Moreover, as for the Cauchy integral theorem, it is sufficient to require that *f* 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 *f* (*z*) = 1/*z*, defined for |*z*| = 1, 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 to construct the holomorphic function from the real part on the boundary. For example, the function *f* (*z*) = *i* − *iz* has real part Re *f* (*z*) = Im *z*. On the unit circle this can be written *i*/*z* − *iz*/2. Using the Möbius transformation and the Stieltjes formula we construct the function inside the circle. The *i*/*z* term makes no contribution, and we find the function −*iz*. This has the correct real part on the boundary, and also gives us the corresponding imaginary part, but off by a constant, namely *i*.

By using the Cauchy integral theorem, one can show that the integral over *C* (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around *a*. Since *f* (*z*) is continuous, we can choose a circle small enough on which *f* (*z*) is arbitrarily close to *f* (*a*). On the other hand, the integral

over any circle *C* centered at *a*. This can be calculated directly via a parametrization (integration by substitution) *z*(*t*) = *a* + *εe ^{it}* where 0 ≤

Letting *ε* → 0 gives the desired estimate

Let

and let *C* be the contour described by |*z*| = 2 (the circle of radius 2).

To find the integral of *g*(*z*) around the contour *C*, we need to know the singularities of *g*(*z*). Observe that we can rewrite *g* as follows:

where *z*_{1} = −1 + *i* and *z*_{2} = −1 − *i*.

Thus, *g* has poles at *z*_{1} and *z*_{2}. 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; that is, we can express the integral around the contour as the sum of the integral around *z*_{1} and *z*_{2} where the contour is a small circle around each pole. Call these contours *C*_{1} around *z*_{1} and *C*_{2} around *z*_{2}.

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 *C*_{1}, define *f*_{1} as *f*_{1}(*z*) = (*z* − *z*_{1})*g*(*z*). This is analytic (since the contour does not contain the other singularity). We can simplify *f*_{1} to be:

and now

Since the Cauchy integral theorem says that:

we can evaluate the integral as follows:

Doing likewise for the other contour:

we evaluate

The integral around the original contour *C* then is the sum of these two integrals:

An elementary trick using partial fraction decomposition:

The integral formula has broad applications. First, it implies that a function which 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 applied to

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 which is not the limit of the derivatives of the members of the sequence.

Another consequence is that if *f* (*z*) = ∑ *a*_{n}*z*^{n} is holomorphic in |*z*| < *R* and 0 < *r* < *R* then the coefficients *a*_{n} satisfy **Cauchy's inequality**^{ [1] }

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

A version of Cauchy's integral formula is the Cauchy–Pompeiu formula,^{ [2] } and holds for smooth functions as well, as it is based on Stokes' theorem. Let *D* be a disc in **C** and suppose that *f* is a complex-valued *C*^{1} function on the closure of *D*. Then^{ [3] }( Hörmander 1966 , Theorem 1.2.1)

One may use this representation formula to solve the inhomogeneous Cauchy–Riemann equations in *D*. Indeed, if *φ* is a function in *D*, then a particular solution *f* of the equation is a holomorphic function outside the support of *μ*. Moreover, if in an open set *D*,

for some *φ* ∈ *C*^{k}(*D*) (where *k* ≥ 1), then *f* (*ζ*, *ζ*) is also in *C*^{k}(*D*) and satisfies the equation

The first conclusion is, succinctly, that the convolution *μ* ∗ *k*(*z*) of a compactly supported measure with the **Cauchy kernel**

is a holomorphic function off the support of *μ*. Here p.v. denotes the principal value. The second conclusion asserts that the Cauchy kernel is a fundamental solution of the Cauchy–Riemann equations. Note that for smooth complex-valued functions *f* of compact support on **C** the generalized Cauchy integral formula simplifies to

and is a restatement of the fact that, considered as a distribution, (π*z*)^{−1} is a fundamental solution of the Cauchy–Riemann operator ∂/∂*z̄*.^{ [4] } The generalized Cauchy integral formula can be deduced for any bounded open region *X* with *C*^{1} boundary ∂*X* from this result and the formula for the distributional derivative of the characteristic function *χ*_{X} of *X*:

where the distribution on the right hand side denotes contour integration along ∂*X*.^{ [5] }

In several complex variables, the Cauchy integral formula can be generalized to polydiscs ( Hörmander 1966 , Theorem 2.2.1). Let *D* be the polydisc given as the Cartesian product of *n* open discs *D*_{1}, ..., *D*_{n}:

Suppose that *f* is a holomorphic function in *D* continuous on the closure of *D*. Then

where *ζ* = (*ζ*_{1},...,*ζ*_{n}) ∈ *D*.

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.

Geometric calculus defines a derivative operator ∇ = **ê**_{i} ∂_{i} under its geometric product — that is, for a *k*-vector field *ψ*(**r**), the derivative ∇*ψ* generally contains terms of grade *k* + 1 and *k* − 1. For example, a vector field (*k* = 1) generally has in its derivative a scalar part, the divergence (*k* = 0), and a bivector part, the curl (*k* = 2). This particular derivative operator has a Green's function:

where *S _{n}* is the surface area of a unit

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

where, for an *n*-dimensional vector space, *d***S** is an (*n* − 1)-vector and *d***V** is an *n*-vector. The function *f* (**r**) can, in principle, be composed of any combination of multivectors. The proof of Cauchy's integral theorem for higher dimensional spaces relies on the using the generalized Stokes theorem on the quantity *G*(**r**, **r**′) *f* (**r**′) and use of the product rule:

When ∇ *f* = 0, *f* (**r**) is called a *monogenic function*, the generalization of holomorphic functions to higher-dimensional spaces — indeed, it can be shown that the Cauchy–Riemann condition is just the two-dimensional expression of the monogenic condition. When that condition is met, the second term in the right-hand integral vanishes, leaving only

where *i _{n}* is that algebra's unit

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.

- Cauchy–Riemann equations
- Methods of contour integration
- Nachbin's theorem
- Morera's theorem
- Mittag-Leffler's theorem
- Green's function generalizes this idea to the non-linear setup
- Schwarz integral formula
- Parseval–Gutzmer formula
- Bochner–Martinelli formula

- ↑ Titchmarsh 1939 , p. 84
- ↑ Pompeiu, D. (1905). "Sur la continuité des fonctions de variables complexes" (PDF).
*Annales de la Faculté des Sciences de Toulouse*.**2**(7.3): 265–315. - ↑ http://people.math.carleton.ca/~ckfong/S32.pdf
- ↑ Hörmander 1983 , pp. 63, 81
- ↑ Hörmander 1983 , pp. 62–63

In mathematics, a **holomorphic function** is a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighborhood of the point. The existence of a complex derivative in a neighbourhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal, locally, to its own Taylor series (*analytic*). Holomorphic functions are the central objects of study in complex analysis.

In number theory, the **prime number theorem** (**PNT**) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann.

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.

In mathematics, the **Laurent series** of a complex function *f*(*z*) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.

In mathematics, more specifically complex analysis, the **residue** is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.

In complex analysis, a discipline within mathematics, the **residue theorem**, sometimes called **Cauchy's residue theorem**, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it can be seen as a special case of the generalized Stokes' theorem.

In vector calculus, **Green's theorem** relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem.

In complex analysis, **Liouville's theorem**, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all in is constant. Equivalently, non-constant holomorphic functions on have unbounded images.

The theory of **functions of several complex variables** is the branch of mathematics dealing with complex-valued functions. The function on the complex coordinate space of n-tuples of complex numbers.

In complex analysis, a branch of mathematics, **Morera's theorem**, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.

In complex analysis a complex-valued function *ƒ* of a complex variable *z*:

In the mathematical field of complex analysis, **contour integration** is a method of evaluating certain integrals along paths in the complex plane.

In complex analysis, the **argument principle** relates the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.

In mathematics, **holomorphic functional calculus** is functional calculus with holomorphic functions. That is to say, given a holomorphic function *f* of a complex argument *z* and an operator *T*, the aim is to construct an operator, *f*(*T*), which naturally extends the function *f* from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of *T* to the bounded operators.

In complex analysis, **Jordan's lemma** is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. It is named after the French mathematician Camille Jordan.

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*. More precisely, given an open set in the complex plane and a function the antiderivative of is a function that satisfies .

In mathematics, a **line integral** is an integral where the function to be integrated is evaluated along a curve. The terms *path integral*, *curve integral*, and *curvilinear integral* are also used; *contour integral* is used as well, although that is typically reserved for line integrals in the complex plane.

In mathematics, **singular integral operators of convolution type** are the singular integral operators that arise on **R**^{n} and **T**^{n} through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on *L*^{2} is evident because the Fourier transform converts them into multiplication operators. Continuity on *L ^{p}* spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on

In mathematics, **singular integral operators on closed curves** arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth Jordan curve in the complex plane and are related by a simple algebraic formula. In the special case of Fourier series for the unit circle, the operators become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure. In general the Cauchy transform is a non-self-adjoint idempotent and the Hilbert transform a non-orthogonal complex structure. The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan curve. The theory for the original curve can be deduced from that of the unit circle, where, because of rotational symmetry, both operators are classical singular integral operators of convolution type. The Hilbert transform satisfies the jump relations of Plemelj and Sokhotski, which express the original function as the difference between the boundary values of holomorphic functions on the region and its complement. Singular integral operators have been studied on various classes of functions, including Hőlder spaces, L^{p} spaces and Sobolev spaces. In the case of L^{2} spaces—the case treated in detail below—other operators associated with the closed curve, such as the Szegő projection onto Hardy space and the Neumann–Poincaré operator, can be expressed in terms of the Cauchy transform and its adjoint.

In mathematics, the **Abel–Plana formula** is a summation formula discovered independently by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820). It states that

- Ahlfors, Lars (1979).
*Complex analysis*(3rd ed.). McGraw Hill. ISBN 978-0-07-000657-7.. - Pompeiu, D. (1905). "Sur la continuité des fonctions de variables complexes" (PDF).
*Annales de la Faculté des Sciences de Toulouse*. Série 2.**7**(3): 265–315. - Titchmarsh, E. C. (1939).
*Theory of functions*(2nd ed.). Oxford University Press. - Hörmander, Lars (1966).
*An Introduction to Complex Analysis in Several Variables*. Van Nostrand. - Hörmander, Lars (1983).
*The Analysis of Linear Partial Differential Operators I*. Springer. ISBN 3-540-12104-8. - Doran, Chris; Lasenby, Anthony (2003).
*Geometric Algebra for Physicists*. Cambridge University Press. ISBN 978-0-521-71595-9.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.