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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. It can also be used to obtain integral formulas for all derivatives of a holomorphic function. The analytic significance of Cauchy's formula is that it shows that in complex analysis "differentiation is equivalent to integration": thus complex differentation, like integration, behaves well under uniform limits, a result which is not true in real analysis.

Suppose U is an open subset of the complex plane C, f : U → C is a holomorphic function and the closed disk D = { z : | z − z₀| ≤ r} is completely contained in U. Let C be the circle forming the boundary of D. Then for every a in the interior of D:

where the contour integral is taken counter-clockwise.

The proof of this statement uses the Cauchy integral theorem and similarly only requires f to be complex differentiable. The formula implies that f is actually infinitely differentiable, with

This formula is sometimes referred to as Cauchy's differentiation formula; it is also a consequence of the fact that holomorphic functions are analytic.

The circle C 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.

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 is 2πi. (This can be calculated directly via parametrization (integration by substitution)

where 0 ≤ t ≤ 2π and ε is the radius of the circle.) Letting ε → 0 gives the desired estimate

Surface of the function g(z) = z² / (z² + 2z + 2) and its singularities, with the contours described in the text.

Consider the function

and the contour described by |z| = 2, call it C.

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

where z₁ = − 1 + i, z₂ = − 1 − i.

Clearly the poles become evident, their moduli are less than 2 and thus lie inside the contour and are subject to consideration by the formula. By the Cauchy-Goursat theorem, we can express the integral around the contour as the sum of the integral around z₁ and z₂ where the contour is a small circle around each pole. Call these contours C₁ around z₁ and C₂ around z₂.

Now, around C₁, f is analytic (since the contour does not contain the other singularity), and this allows us to write f in the form we require, viz:

and now

Doing likewise for the other contour:

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

Because one may deduce from the formula that f must be infinitely often continuously differentiable, the Cauchy integral theorem has broad implications. It is used to prove the residue theorem, which is a far-reaching generalization that removes the requirement that the function be analytic in the enclosed region. The Cauchy integral theorem has no counterpart in real analysis because for a real valued function possession of a first derivative by a function will not guarantee the existence of higher order derivatives. In contrast to this, the proof of the Cauchy integral formula for n^th derivatives shows that analytic functions posses derivatives of all orders.^[1]

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.

1. ^ Transform Calculus: with an Introduction to Complex Variables by E. J. Scott

• Cauchy-Riemann equations
• Methods of contour integration
• Nachbin's theorem
• Morera's theorem

• Eric W. Weisstein, Cauchy Integral Formula at MathWorld.
• Cauchy Integral Formula Module by John H. Mathews