Infinitely Differentiable: Derivatives by Integral

From one formula to a whole tower

In the previous guide the Cauchy integral formula handed you something almost too generous: the value of a holomorphic f at any interior point z_0 is recovered from its values on a surrounding loop C, by integral of f(w)/(w - z_0) dw over C, divided by 2 pi i. The function knows itself from its boundary alone. This guide asks the obvious follow-up question and finds an extraordinary answer: what happens if we differentiate that formula with respect to z_0?

Here is the key move, and it is wonderfully simple. The only place z_0 appears on the right-hand side is inside the kernel 1/(w - z_0). To differentiate the whole integral in z_0, we just differentiate that kernel — the integration is over w, the loop is fixed, and z_0 is a free parameter we are allowed to wiggle. Since d/dz_0 of 1/(w - z_0) equals 1/(w - z_0)^2, the derivative f'(z_0) is itself an integral: 1/(2 pi i) times the integral of f(w)/(w - z_0)^2 dw over C. We never touched f; we only nudged the kernel.

The formula for every derivative

Nothing stops us from differentiating again. Each pass through the integral drops one more power onto the kernel and multiplies by the next integer, exactly as differentiating 1/(w - z_0) repeatedly would. After n differentiations the kernel is 1/(w - z_0)^(n+1) and the accumulated constant is n factorial. This is the generalized Cauchy formula, and it is one of the cleanest equations in all of analysis.

                       n!         f(w)
f^(n)(z_0)  =  -------  *  integral over C  -------------- dw
                      2 pi i                 (w - z_0)^(n+1)

n = 0:  f(z_0)    has kernel 1/(w - z_0)        (the original formula)
n = 1:  f'(z_0)   has kernel 1/(w - z_0)^2
n = 2:  f''(z_0)  has kernel 2/(w - z_0)^3 ... and the 2 is the 2! out front

Read the formula slowly, because it makes a claim that should stop you. The right-hand side is built only from the values of f on the loop C — never from any derivative of f. Yet the left-hand side is the n-th derivative. So the integral formula does not merely compute f'; it manufactures f', f'', f''', and every higher derivative purely out of boundary values. The loop on the right does not even need to know that f is differentiable; it only needs the numbers f(w). The differentiation has been quietly outsourced to the kernel.

The miracle: one derivative forces infinitely many

Now collect the harvest. To even write down the Cauchy integral formula, all we ever assumed was that f is holomorphic — that ONE complex derivative exists at each point of the region. But the generalized formula then exhibits f''(z_0), f'''(z_0), and every f^(n)(z_0) as honest, finite integrals. Each one exists. So a function that is complex-differentiable just once is automatically differentiable infinitely many times. This is infinite differentiability, and it is the headline result of the entire rung.

The same machinery quietly delivers more than smoothness. Expand the kernel 1/(w - z_0) as a geometric series in (z_0 - a)/(w - a) about a center a, and integrate term by term: the Cauchy formula reorganizes itself into a power series sum a_n (z_0 - a)^n that converges in a disk. That is the bridge called the Cauchy integral as a route to series, and it is why the very next results of complex analysis — Taylor expansions, Liouville's theorem, the identity theorem — are all downstream of this one formula. Smoothness and analyticity arrive together, from the same integral.

Cauchy estimates: bounding derivatives by the loop

The integral formula is not only an existence statement; it is a measuring instrument. Apply the ML inequality from the previous rung to the generalized formula on a circle of radius r centered at z_0. The kernel has size 1/r^(n+1) on that circle, the length is 2 pi r, and if |f| is bounded by some M there, the constants combine cleanly into the Cauchy estimates: |f^(n)(z_0)| <= n! M / r^n. In particular |f'(z_0)| <= M / r and |f''(z_0)| <= 2 M / r^2, where M is the maximum of |f| on the circle.

These inequalities punch far above their weight. Because they let you bound the derivatives of f at the center using nothing but a bound on f itself out near the boundary, they are the lever behind several giants you will meet next. Let r grow without limit while M stays fixed and you crush f' to zero — that is the heart of Liouville's theorem, that a bounded entire function must be constant, which in turn cracks open the fundamental theorem of algebra. The Cauchy estimates are small, but they are the fulcrum.

What this unlocks, and an honest caveat

Step back and see how much one formula bought. Differentiating the Cauchy integral formula gave a formula for every derivative; that gave infinite differentiability for free; the same expansion gave a convergent power series; and an ML estimate of the same integral gave the Cauchy estimates, the lever for Liouville and the fundamental theorem of algebra. Even the converse direction comes within reach: if every loop integral of a continuous function vanishes, the same parameter-integral machinery builds a primitive and proves the function holomorphic — that is the promise of Morera's theorem, which the fourth guide of this rung makes precise.