Cauchy's integral formula

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

it follows that holomorphic functions are analytic, i.e. they can be expanded as convergent power series.

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.

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.

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.

Using the Möbius transformation and the Stieltjes formula we construct the function inside the circle.

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.

First, it implies that a function which is holomorphic in an open set is in fact infinitely differentiable there.

The proof of this uses the dominated convergence theorem and the geometric series applied to

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.

Another consequence is that if f(z) = Σ an zn is holomorphic in |z| < R and 0 < r < R then the coefficients an satisfy Cauchy's estimate[1]

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

Let D be a disc in C and suppose that f is a complex-valued C1 function on the closure of D. Then[4][5][6]

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

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, (πz)−1 is a fundamental solution of the Cauchy–Riemann operator ⁠∂/∂z̄⁠.

[7] The generalized Cauchy integral formula can be deduced for any bounded open region X with C1 boundary ∂X from this result and the formula for the distributional derivative of the characteristic function χX of X:

of the form "distribution times C∞ function", so we may apply the Leibniz rule to calculate its derivatives: Using that (πz)−1 is a fundamental solution of the Cauchy–Riemann operator ⁠∂/∂z̄⁠, we get

In several complex variables, the Cauchy integral formula can be generalized to polydiscs.

[9] Let D be the polydisc given as the Cartesian product of n open discs D1, ..., Dn:

where ζ = (ζ1,...,ζn) ∈ 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 ∇ = êi ∂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.

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