In mathematics, Cauchy's integral formula, named after
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...
, is a central statement in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
. It expresses the fact that a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
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
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
.
Theorem
Let be an
open subset of the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
, and suppose the closed disk defined as
is completely contained in . Let be a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
, and let be the
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
, oriented
counterclockwise, forming the
boundary of . Then for every in the
interior of ,
The proof of this statement uses the
Cauchy integral theorem
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in t ...
and like that theorem, it only requires to be
complex differentiable
In mathematics, a holomorphic function is a complex-valued function of one or Function of several complex variables, more complex number, complex variables that is Differentiable function#Differentiability in complex analysis, complex differ ...
. 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 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 can be replaced by any closed
rectifiable curve in which has
winding number
In mathematics, the winding number or winding index of a closed curve in the plane (mathematics), plane around a given point (mathematics), point is an integer representing the total number of times that the curve travels counterclockwise aroun ...
one about . Moreover, as for the Cauchy integral theorem, it is sufficient to require that 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 Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation "
* if and are related by , that is,
* if holds, that is,
* if the equivalence classes of and with respect to are equal.
This figure of speech ...
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
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying .
Geometrically ...
and the
Stieltjes inversion formula to construct the holomorphic function from the real part on the boundary. For example, the function has real part . On the unit circle this can be written . Using the Möbius transformation and the Stieltjes formula we construct the function inside the circle. The 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
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in t ...
, one can show that the integral over (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around . Since is continuous, we can choose a circle small enough on which is arbitrarily close to . On the other hand, the integral
over any circle centered at . This can be calculated directly via a parametrization (
integration by substitution
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and c ...
) where and is the radius of the circle.
Letting gives the desired estimate
Example

Let
and let be the contour described by (the circle of radius 2).
To find the integral of around the contour , we need to know the singularities of . Observe that we can rewrite as follows:
where and .
Thus, has poles at 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; that is, we can express the integral around the contour as the sum of the integral around and where the contour is a small circle around each pole. Call these contours around and 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 , define as . This is
analytic (since the contour does not contain the other singularity). We can simplify to be:
and now
Since the Cauchy integral formula says that:
we can evaluate the integral as follows:
Doing likewise for the other contour:
we evaluate
The integral around the original contour then is the sum of these two integrals:
An elementary trick using
partial fraction decomposition:
Consequences
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
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
, meaning that it can be represented as a
power series. The proof of this uses the
dominated convergence theorem and the
geometric series
In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
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 function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that i ...
s; 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 is holomorphic in and then the coefficients 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
In other words, the average value of over the circle centered at with radius 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 function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain.
A function of class C^k is a function of smoothness at least ; t ...
s as well, as it is based on
Stokes' theorem. Let be a disc in and suppose that is a complex-valued function on the
closure of . Then
One may use this representation formula to solve the inhomogeneous
Cauchy–Riemann equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin-Louis Cauchy, Augustin Cauchy and Bernhard Riemann, consist of a system of differential equations, system of two partial differential equatio ...
in . Indeed, if is a function in , then a particular solution of the equation is a holomorphic function outside the support of . Moreover, if in an open set ,
for some (where ), then is also in and satisfies the equation
The first conclusion is, succinctly, that the
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of a compactly supported measure with the Cauchy kernel
is a holomorphic function off the support of . Here denotes the
principal value. The second conclusion asserts that the Cauchy kernel is a
fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ...
of the Cauchy–Riemann equations. Note that for smooth complex-valued functions of compact support on the generalized Cauchy integral formula simplifies to
and is a restatement of the fact that, considered as a
distribution, is a
fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ...
of the
Cauchy–Riemann operator .
The generalized Cauchy integral formula can be deduced for any bounded open region with boundary 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
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
Contour integration is closely related to the Residue theorem, calculus of residues, a method of co ...
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.
Let be the polydisc given as the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
of open discs :
Suppose that is a holomorphic function in continuous on the closure of . Then
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.
Geometric calculus defines a derivative operator under its geometric product — that is, for a -vector field , the derivative generally contains terms of grade and . For example, a vector field () generally has in its derivative a scalar part, the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
(), and a bivector part, the
curl (). This particular derivative operator has a
Green's function:
where is the surface area of a unit -
ball
A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ...
in the space (that is, , the circumference of a circle with radius 1, and , the surface area of a sphere with radius 1). By definition of a Green's function,
It is this useful property that can be used, in conjunction with the generalized Stokes theorem:
where, for an -dimensional vector space, is an -vector and is an -vector. The function 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 and use of the product rule:
When , 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 is that algebra's unit -vector, the
pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.
A pseudoscalar, when multiplied by an ordinary vector, becomes a '' pseudovector'' ...
. The result is
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
*
Cauchy–Riemann equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin-Louis Cauchy, Augustin Cauchy and Bernhard Riemann, consist of a system of differential equations, system of two partial differential equatio ...
*
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
*
Helffer–Sjöstrand formula
Notes
References
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External links
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{{DEFAULTSORT:Cauchy's Integral Formula
Augustin-Louis Cauchy
Theorems in complex analysis