Derivatives from Integrals

Complex Analysis · Axiom Academy

LESSON Computing Derivatives from Integrals Differentiate Cauchy's formula once and you get every derivative of an analytic function — as a contour integral. 1. Differentiating Under the Integral Cauchy's formula writes f(a) as an integral around a contour C that encloses the point a . To get f'(a) , differentiate both sides with respect to a . The values f(z) on C don't depend on a at all — only the factor does. So only the denominator gets differentiated , and one extra power appears. The value: a single power of z-a downstairs The derivative: squares the power 2. The Pattern: Powers and Factorials Differentiate again, and again. Each pass does two things in lockstep: it raises the power in the denominator by one, and it pulls down the next whole number as a factor. After n differentiations the denominator is (z-a)^ n+1 and the accumulated factor is n! — giving one master formula for every derivative. Power (z-a)^ 1 , factor 0!=1 . This is Cauchy's formula for the value f(a) . Power (z-a)^ 2 , factor 1!=1 . The first-derivative formula from Step 1. Power (z-a)^ 3 , factor 2!=2 . The squaring becomes cubing; the 2 shows up. Power (z-a)^ n+1 , factor n! . One formula covers the whole tower of derivatives. To match an integral to this formula, count the power downstairs. A denominator of (z-a)^ n+1 means order n ; the integral then equals — the derivative is hiding in plain sight. 3. One Derivative Unlocks Them All

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