Cauchy's integral formula and derivative estimates

Suppose f is holomorphic on an open neighborhood of a closed disk centered at a with radius R. With counterclockwise boundary orientation, its value at any interior point is the integral of f divided by the displacement from that point, scaled by 1/(2πi). The kernel has a singularity at the evaluation point, but the original f is holomorphic there. One proof removes a tiny circle around that point, applies Cauchy's theorem to the intervening region, and evaluates the limiting small-circle contribution.

Differentiating with respect to the interior evaluation point yields formulas for all higher derivatives. The contour remains a positive distance from a compact interior neighborhood, so the kernel derivatives are uniformly controlled and differentiation under the integral is justified. This establishes that a holomorphic function has derivatives of every order, rather than assuming such smoothness at the outset. Reversing orientation reverses the integral; the standard displayed formula always uses positive orientation.

Let M bound the modulus of f on the circle. The contour length is 2πR and the denominator has constant modulus $R^{m+1}$. Applying the length bound to Cauchy's formula gives the derivative estimate below. Notice that larger available circles improve derivative control only when their boundary maxima do not grow too quickly. For an entire function that is globally bounded by one constant, the first-derivative estimate tends to zero as R tends to infinity at every center.

That proves Liouville's theorem: a bounded entire function is constant. Being bounded on a single disk is not enough, and being bounded only along the real axis is not enough. For example, a complex exponential can remain bounded or oscillatory along selected directions while growing along others. The estimate measures a whole surrounding circle, so it explicitly prevents replacing a complex-domain hypothesis with a convenient real slice.

For points inside a smaller concentric circle, expand the Cauchy kernel as a geometric series. Its convergence is uniform along the fixed integration circle, so term-by-term integration is legitimate. The coefficients become the derivatives at the center divided by factorials. This proves a local Taylor representation and explains why holomorphicity is equivalent to local analyticity. A smooth real function need not equal its Taylor series, so that equivalence is a special consequence of complex differentiability.

A practical calculation separates exact evaluation from an estimate. If the numerator's derivative at the center is known, the formula gives an exact contour integral. If only a modulus bound is available, it gives an inequality and cannot identify the actual derivative. The largest usable circle is constrained by the function's domain and singularities, not by the desire for a small bound. This distinction will reappear when Taylor series stop at nearby singularities and require analytic continuation beyond their original disks.