Computing Higher Derivatives

Complex Analysis · Axiom Academy

EXAMPLE Computing Higher Derivatives via Cauchy's Formula Find a second derivative by integrating around a contour — no differentiation required The function f(z) = e^ z is analytic on the whole plane. Use the Cauchy Integral Formula for derivatives to compute the second derivative f''(0) from a contour integral around the unit circle. You computed a second derivative without differentiating once — the value fell out of a single contour integral. Cauchy's derivative formula: for f analytic inside and on C and z_0 inside, . The order sets the power: the n -th derivative puts (w-z_0)^ n+1 in the denominator — for n=2 that is the cube w^3 , not w^2 . The residue does the work: is the coefficient of in the Laurent series, which is , so the integral is . Result: — exactly e^ 0 , the answer plain calculus would give. Turning differentiation into integration is what makes analytic functions so rigid: every derivative at a point is already encoded in the function's values around any loop enclosing it.

This is the written version of the interactive lesson above. See the full Complex Analysis course.