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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
One loop holds every inside value and every derivative, and bounded everywhere forces constant.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.