Provides integral formulas for all derivatives of a holomorphic function
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 "U" be an open subset of the complex plane C, and suppose the closed disk "D" defined as
formula_1
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",
formula_2
The proof of this statement uses the Cauchy integral theorem and like that theorem, it only requires "f" to be complex differentiable. Since formula_3 can be expanded as a power series in the variable formula_4
formula_5
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
formula_6
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") 
 , 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 . 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 −"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".
Proof sketch.
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
formula_7
over any circle "C" centered at "a". This can be calculated directly via a parametrization (integration by substitution) "z"("t") 
 "a" + "εeit" where 0 ≤ "t" ≤ 2π and "ε" is the radius of the circle.
Letting "ε" → 0 gives the desired estimate
formula_8
Example.
Let
formula_9
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:
formula_10
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:
formula_11
and now
formula_12
Since the Cauchy integral formula says that:
formula_13
we can evaluate the integral as follows:
formula_14
Doing likewise for the other contour:
formula_15
we evaluate
formula_16
The integral around the original contour "C" then is the sum of these two integrals:
formula_17
An elementary trick using partial fraction decomposition:
formula_18
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
formula_19
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") = Σ "a""n" "z""n" is holomorphic in and 0 ; "r" ; "R" then the coefficients "a""n" satisfy Cauchy's estimate
formula_20
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
formula_21
In other words, the average value of "f" over the circle centered at "z" with radius "r" is "f"("z"). 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. Let "D" be a disc in C and suppose that "f" is a complex-valued "C"1 function on the closure of "D". Then
formula_22
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",
formula_23
for some "φ" ∈ "C""k"("D") (where "k" ≥ 1), then "f"("ζ", "ζ") is also in "C""k"("D") and satisfies the equation
formula_24
The first conclusion is, succinctly, that the convolution "μ" ∗ "k"("z") of a compactly supported measure with the Cauchy kernel
formula_25
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
formula_26
and is a restatement of the fact that, considered as a distribution, (π"z")−1 is a fundamental solution of the Cauchy–Riemann operator .
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":
formula_27
where the distribution on the right hand side denotes contour integration along ∂"X".
 ;templatestyles src="Math_proof/styles.css" / ;Proof
For formula_28 calculate:
formula_29
then traverse formula_30 in the anti-clockwise direction. Fix a point formula_31 and let formula_32 denote arc length on formula_30 measured from formula_34 anti-clockwise. Then, if formula_35 is the length of formula_36 is a parametrization of formula_30. The derivative formula_38 is a unit tangent to formula_30 and formula_40 is the unit outward normal on formula_30. We are lined up for use of the divergence theorem: put formula_42 so that formula_43 and we get
formula_44
Hence we proved formula_45.
Now we can deduce the generalized Cauchy integral formula:
 ;templatestyles src="Math_proof/styles.css" / ;Proof
^1(X) ;/math ; and since formula_46 this distribution is locally in formula_47 of the form "distribution times C∞ function", so we may apply the Leibniz rule to calculate its derivatives:
formula_48
Using that (π"z")−1 is a fundamental solution of the Cauchy–Riemann operator , we get formula_49:
formula_50
Applying formula_51 to formula_52:
formula_53
where formula_45 is used in the last line.
Rearranging, we get
formula_55
as desired.
Several variables.
In several complex variables, the Cauchy integral formula can be generalized to polydiscs. Let "D" be the polydisc given as the Cartesian product of "n" open discs "D"1, ..., "D""n":
formula_56
Suppose that "f" is a holomorphic function in "D" continuous on the closure of "D". Then
formula_57
where "ζ" = ("ζ"1...,"ζ""n") ∈ "D".
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 ∇ = 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:
formula_58
where "Sn" is the surface area of a unit "n"-ball in the space (that is, "S"2 = 2π, the circumference of a circle with radius 1, and "S"3 = 4π, the surface area of a sphere with radius 1). By definition of a Green's function,
formula_59
It is this useful property that can be used, in conjunction with the generalized Stokes theorem:
formula_60
where, for an "n"-dimensional vector space, "dS is an ("n" − 1)-vector and "dV 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:
formula_61
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
formula_62
where "in" is that algebra's unit "n"-vector, the pseudoscalar. The result is
formula_63
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.
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Notes.
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References.
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