---
title: "Cauchy's Integral Formula and Derivatives of Every Order"
description: "Recover a function's value inside a contour from its values on the contour, then differentiate under the integral to get every derivative at once."
canonical: https://lightmysky.com/learn/mathematics/cauchys-integral-formula-and-derivatives-of-every-order-mt_mViFwrFNWQ
source: https://lightmysky.com/learn/mathematics/cauchys-integral-formula-and-derivatives-of-every-order-mt_mViFwrFNWQ.md
retrieved: 2026-09-12
---

> **Agent view.** This is the Markdown twin of the page, for tools and assistants.
> When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md
> API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md
> Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt
> Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json
> Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`.

# Cauchy's Integral Formula and Derivatives of Every Order

Recover a function's value inside a contour from its values on the contour, then differentiate under the integral to get every derivative at once.

Subject: Mathematics · Area: Complex Analysis · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/cauchys-integral-formula-and-derivatives-of-every-order-mt_mViFwrFNWQ

## Ready when they can

- State and apply the integral formula to evaluate a value or a derivative
- Explain why complex differentiability once forces differentiability infinitely often
- Deduce Liouville's theorem and, from it, the fundamental theorem of algebra

## Lesson: One loop that rebuilds a whole function

Cauchy's integral formula rebuilds inside values from boundary values. For f analytic inside and on a loop, f of a equals 1 over 2 pi i times the loop integral of f of z over (z minus a). On the circle of radius 2 the point 1 sits inside, so the formula returns f of 1. The standard positive unit circle winds once, giving winding number 1.

**Example.** Put a equal 0.5 inside the unit circle. The integral of f of z over (z minus 0.5) around it equals 2 pi i times f of 0.5. The undivided integral keeps its 2 pi i factor; dividing by it leaves exactly the inside value.

Differentiate under the integral sign and the same loop delivers every derivative, since a appears only in the smooth denominator. Complex differentiability once forces derivatives of all orders. Real functions offer no such promise: once differentiable can still be rough at the next level.

**Tip.** Derivative estimates give Liouville: a bounded entire function must be constant, since the bound on f prime shrinks like 1 over R. The fundamental theorem of algebra follows: a zero free polynomial would have a bounded reciprocal, contradicting Liouville. Complex sine is no counterexample, since it grows along the imaginary axis.

**Recap.** One loop holds every inside value and every derivative, and bounded everywhere forces constant.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Cauchy's Theorem and Deforming a Contour](https://lightmysky.com/learn/mathematics/cauchys-theorem-and-deforming-a-contour-mt_aacYFVN-HH)
- [Roots of Polynomials and Roots of Unity](https://lightmysky.com/learn/mathematics/roots-of-polynomials-and-roots-of-unity-mt_XAISygqdQr)

## Opens up

- [Power Series and Analyticity in the Complex Plane](https://lightmysky.com/learn/mathematics/power-series-and-analyticity-in-the-complex-plane-mt_Mwxapy_pU8)