Английская Википедия:Cauchy's integral formula
Шаблон:Short description Шаблон:Distinguish Шаблон:Complex analysis sidebar 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
Let Шаблон:Math be an open subset of the complex plane Шаблон:Math, and suppose the closed disk Шаблон:Math defined as <math display="block">D = \bigl\{z:|z - z_0| \leq r\bigr\}</math> is completely contained in Шаблон:Math. Let Шаблон:Math be a holomorphic function, and let Шаблон:Math be the circle, oriented counterclockwise, forming the boundary of Шаблон:Math. Then for every Шаблон:Math in the interior of Шаблон:Math, <math display="block">f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,dz.\,</math>
The proof of this statement uses the Cauchy integral theorem and like that theorem, it only requires Шаблон:Math to be complex differentiable. Since <math>1/(z-a)</math> can be expanded as a power series in the variable <math>a</math> <math display="block">\frac{1}{z-a} = \frac{1+\frac{a}{z}+\left(\frac{a}{z}\right)^2+\cdots}{z}</math> it follows that holomorphic functions are analytic, i.e. they can be expanded as convergent power series. In particular Шаблон:Math is actually infinitely differentiable, with <math display="block">f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\,dz.</math>
This formula is sometimes referred to as Cauchy's differentiation formula.
The theorem stated above can be generalized. The circle Шаблон:Math can be replaced by any closed rectifiable curve in Шаблон:Math which has winding number one about Шаблон:Math. Moreover, as for the Cauchy integral theorem, it is sufficient to require that Шаблон:Math 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 Шаблон:Math, defined for Шаблон:Math, 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 Шаблон:Math has real part Шаблон:Math. On the unit circle this can be written Шаблон:Math. Using the Möbius transformation and the Stieltjes formula we construct the function inside the circle. The Шаблон:Math term makes no contribution, and we find the function Шаблон:Math. This has the correct real part on the boundary, and also gives us the corresponding imaginary part, but off by a constant, namely Шаблон:Math.
Proof sketch
By using the Cauchy integral theorem, one can show that the integral over Шаблон:Math (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around Шаблон:Math. Since Шаблон:Math is continuous, we can choose a circle small enough on which Шаблон:Math is arbitrarily close to Шаблон:Math. On the other hand, the integral <math display="block">\oint_C \frac{1}{z-a} \,dz = 2 \pi i,</math> over any circle Шаблон:Math centered at Шаблон:Math. This can be calculated directly via a parametrization (integration by substitution) Шаблон:Math where Шаблон:Math and Шаблон:Math is the radius of the circle.
Letting Шаблон:Math gives the desired estimate <math display="block">\begin{align} \left | \frac{1}{2 \pi i} \oint_C \frac{f(z)}{z-a} \,dz - f(a) \right | &= \left | \frac{1}{2 \pi i} \oint_C \frac{f(z)-f(a)}{z-a} \,dz \right | \\[1ex] &= \left | \frac{1}{2\pi i}\int_0^{2\pi}\left(\frac{f\bigl(z(t)\bigr)-f(a)}{\varepsilon e^{it}}\cdot\varepsilon e^{it} i\right )\,dt\right | \\[1ex] &\leq \frac{1}{2 \pi} \int_0^{2\pi} \frac{ \left|f\bigl(z(t)\bigr) - f(a)\right| } {\varepsilon} \,\varepsilon\,dt \\[1ex] &\leq \max_{|z-a|=\varepsilon} \left|f(z) - f(a)\right| ~~ \xrightarrow[\varepsilon\to 0]{} ~~ 0. \end{align}</math>
Example
Let <math display="block">g(z) = \frac{z^2}{z^2+2z+2},</math> and let Шаблон:Math be the contour described by Шаблон:Math (the circle of radius 2).
To find the integral of Шаблон:Math around the contour Шаблон:Math, we need to know the singularities of Шаблон:Math. Observe that we can rewrite Шаблон:Math as follows: <math display="block">g(z) = \frac{z^2}{(z-z_1)(z-z_2)}</math> where Шаблон:Math and Шаблон:Math.
Thus, Шаблон:Math has poles at Шаблон:Math and Шаблон:Math. 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 Шаблон:Math and Шаблон:Math where the contour is a small circle around each pole. Call these contours Шаблон:Math around Шаблон:Math and Шаблон:Math around Шаблон:Math.
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 Шаблон:Math, define Шаблон:Math as Шаблон:Math. This is analytic (since the contour does not contain the other singularity). We can simplify Шаблон:Math to be: <math display="block">f_1(z) = \frac{z^2}{z-z_2}</math> and now <math display="block">g(z) = \frac{f_1(z)}{z-z_1}.</math>
Since the Cauchy integral formula says that: <math display="block">\oint_C \frac{f_1(z)}{z-a}\, dz=2\pi i\cdot f_1(a),</math> we can evaluate the integral as follows: <math display="block">
\oint_{C_1} g(z)\,dz
=\oint_{C_1} \frac{f_1(z)}{z-z_1}\,dz
=2\pi i\frac{z_1^2}{z_1-z_2}.
</math>
Doing likewise for the other contour: <math display="block">f_2(z) = \frac{z^2}{z-z_1},</math> we evaluate <math display="block">
\oint_{C_2} g(z)\,dz
=\oint_{C_2} \frac{f_2(z)}{z-z_2}\,dz
=2\pi i\frac{z_2^2}{z_2-z_1}.
</math>
The integral around the original contour Шаблон:Math then is the sum of these two integrals: <math display="block">\begin{align}
\oint_C g(z)\,dz
&{}= \oint_{C_1} g(z)\,dz
+ \oint_{C_2} g(z)\,dz \\[.5em]
&{}= 2\pi i\left(\frac{z_1^2}{z_1-z_2}+\frac{z_2^2}{z_2-z_1}\right) \\[.5em] &{}= 2\pi i(-2) \\[.3em] &{}=-4\pi i. \end{align}</math>
An elementary trick using partial fraction decomposition: <math display="block">
\oint_C g(z)\,dz
=\oint_C \left(1-\frac{1}{z-z_1}-\frac{1}{z-z_2}\right) \, dz
=0-2\pi i-2\pi i
=-4\pi i
</math>
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, 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
<math display="block">f(\zeta) = \frac{1}{2\pi i}\int_C \frac{f(z)}{z-\zeta}\,dz.</math>
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 Шаблон:Math is holomorphic in Шаблон:Math and Шаблон:Math then the coefficients Шаблон:Math satisfy Cauchy's inequality[1] <math display="block">|a_n|\le r^{-n} \sup_{|z|=r}|f(z)|.</math>
From Cauchy's inequality, 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[2] <math display="block">f(z) = \frac{1}{2\pi} \int_{0}^{2\pi} f(z + r e^{i\theta}) \, d\theta.</math>
In other words, the average value of Шаблон:Math over the circle centered at Шаблон:Math with radius Шаблон:Math is Шаблон:Math. 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,[3] and holds for smooth functions as well, as it is based on Stokes' theorem. Let Шаблон:Math be a disc in Шаблон:Math and suppose that Шаблон:Math is a complex-valued Шаблон:Math function on the closure of Шаблон:Math. Then[4][5] <math display="block">f(\zeta) = \frac{1}{2\pi i}\int_{\partial D} \frac{f(z) \,dz}{z-\zeta} - \frac{1}{\pi}\iint_D \frac{\partial f}{\partial \bar{z}}(z) \frac{dx\wedge dy}{z-\zeta}.</math>
One may use this representation formula to solve the inhomogeneous Cauchy–Riemann equations in Шаблон:Math. Indeed, if Шаблон:Math is a function in Шаблон:Math, then a particular solution Шаблон:Math of the equation is a holomorphic function outside the support of Шаблон:Math. Moreover, if in an open set Шаблон:Math, <math display="block">d\mu = \frac{1}{2\pi i}\varphi \, dz\wedge d\bar{z}</math> for some Шаблон:Math (where Шаблон:Math), then Шаблон:Math is also in Шаблон:Math and satisfies the equation <math display="block">\frac{\partial f}{\partial\bar{z}} = \varphi(z,\bar{z}).</math>
The first conclusion is, succinctly, that the convolution Шаблон:Math of a compactly supported measure with the Cauchy kernel <math display="block">k(z) = \operatorname{p.v.}\frac{1}{z}</math> is a holomorphic function off the support of Шаблон:Math. Here Шаблон:Math 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 Шаблон:Math of compact support on Шаблон:Math the generalized Cauchy integral formula simplifies to <math display="block">f(\zeta) = \frac{1}{2\pi i}\iint \frac{\partial f}{\partial \bar{z}}\frac{dz\wedge d\bar{z}}{z-\zeta},</math> and is a restatement of the fact that, considered as a distribution, Шаблон:Math is a fundamental solution of the Cauchy–Riemann operator Шаблон:Math.[6] The generalized Cauchy integral formula can be deduced for any bounded open region Шаблон:Math with Шаблон:Math boundary Шаблон:Math from this result and the formula for the distributional derivative of the characteristic function Шаблон:Math of Шаблон:Math: <math display="block"> \frac {\partial \chi_X}{\partial \bar z}= \frac{i}{2} \oint_{\partial X} \,dz,</math> where the distribution on the right hand side denotes contour integration along Шаблон:Math.[7]
Several variables
In several complex variables, the Cauchy integral formula can be generalized to polydiscs.[8] Let Шаблон:Math be the polydisc given as the Cartesian product of Шаблон:Math open discs Шаблон:Math: <math display="block">D = \prod_{i=1}^n D_i.</math>
Suppose that Шаблон:Math is a holomorphic function in Шаблон:Math continuous on the closure of Шаблон:Math. Then <math display="block">f(\zeta) = \frac{1}{\left(2\pi i\right)^n}\int\cdots\iint_{\partial D_1\times\cdots\times\partial D_n} \frac{f(z_1,\ldots,z_n)}{(z_1-\zeta_1)\cdots(z_n-\zeta_n)} \, dz_1\cdots dz_n</math> where Шаблон:Math.
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 Шаблон:Math under its geometric product — that is, for a Шаблон:Math-vector field Шаблон:Math, the derivative Шаблон:Math generally contains terms of grade Шаблон:Math and Шаблон:Math. For example, a vector field (Шаблон:Math) generally has in its derivative a scalar part, the divergence (Шаблон:Math), and a bivector part, the curl (Шаблон:Math). This particular derivative operator has a Green's function: <math display="block">G\left(\mathbf r, \mathbf r'\right) = \frac{1}{S_n} \frac{\mathbf r - \mathbf r'}{\left|\mathbf r - \mathbf r'\right|^n}</math> where Шаблон:Math is the surface area of a unit Шаблон:Math-ball in the space (that is, Шаблон:Math, the circumference of a circle with radius 1, and Шаблон:Math, the surface area of a sphere with radius 1). By definition of a Green's function, <math display="block">\nabla G\left(\mathbf r, \mathbf r'\right) = \delta\left(\mathbf r- \mathbf r'\right).</math>
It is this useful property that can be used, in conjunction with the generalized Stokes theorem: <math display="block">\oint_{\partial V} d\mathbf S \; f(\mathbf r) = \int_V d\mathbf V \; \nabla f(\mathbf r)</math> where, for an Шаблон:Math-dimensional vector space, Шаблон:Math is an Шаблон:Math-vector and Шаблон:Math is an Шаблон:Math-vector. The function Шаблон:Math 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 Шаблон:Math and use of the product rule: <math display="block">\oint_{\partial V'} G\left(\mathbf r, \mathbf r'\right)\; d\mathbf S' \; f\left(\mathbf r'\right) = \int_V \left(\left[\nabla' G\left(\mathbf r, \mathbf r'\right)\right] f\left(\mathbf r'\right) + G\left(\mathbf r, \mathbf r'\right) \nabla' f\left(\mathbf r'\right)\right) \; d\mathbf V</math>
When Шаблон:Math, Шаблон:Math 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 <math display="block">\oint_{\partial V'} G\left(\mathbf r, \mathbf r'\right)\; d\mathbf S' \; f\left(\mathbf r'\right) = \int_V \left[\nabla' G\left(\mathbf r, \mathbf r'\right)\right] f\left(\mathbf r'\right) = -\int_V \delta\left(\mathbf r - \mathbf r'\right) f\left(\mathbf r'\right) \; d\mathbf V =- i_n f(\mathbf r)</math> where Шаблон:Math is that algebra's unit Шаблон:Math-vector, the pseudoscalar. The result is <math display="block">f(\mathbf r) =- \frac{1}{i_n} \oint_{\partial V} G\left(\mathbf r, \mathbf r'\right)\; d\mathbf S \; f\left(\mathbf r'\right) = -\frac{1}{i_n} \oint_{\partial V} \frac{\mathbf r - \mathbf r'}{S_n \left|\mathbf r - \mathbf r'\right|^n} \; d\mathbf S \; f\left(\mathbf r'\right)</math>
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
- 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
Notes
References
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
External links
