The proof of this statement uses the Cauchy integral theorem and similarly only requires f to be complex differentiable. Since the denominator of the integrand in Cauchy's integral formula can be expanded as a power series in the variable (a - z0), it follows that holomorphic functions are analytic. In particular f is actually infinitely differentiable, with
This formula is sometimes referred to as Cauchy's differentiation formula.
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
Consider the function
and the contour described by |z| = 2, call it C.
To find 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:
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, namely:
Doing likewise for the other contour:
The integral around the original contour C then is the sum of these two integrals:
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.
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 given by
is a holomorphic function outside the support of μ. Moreover, if in an open set D,
for some φ ∈ Ck(D) (k≥1), then is also in Ck(D) and satisfies the equation
The first conclusion is, succinctly, that the convolution μ*k(z) of a compactly supported measure with the Cauchy kernel
is a holomorphic function off the support of μ. The second conclusion asserts that the Cauchy kernel is a fundamental solution of the Cauchy-Riemann equations.
where ζ=(ζ1,...,ζn) ∈ D.
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