Using Antiderivatives

Complex Analysis · Axiom Academy

Evaluating a contour integral with the Fundamental Theorem for analytic functions Evaluate the contour integral , where C is the straight line segment from 0 to 1+i . The contour C (solid) runs straight from 0 to 1+i . Because z^2 is analytic, the dashed corner path gives the same value — the integral depends only on the endpoints. Nicely done. You evaluated a contour integral without ever parametrizing the path — just by finding an antiderivative. The shortcut: if F'(z)=f(z) on a domain containing C , then , exactly like the real Fundamental Theorem of Calculus. Power rule carries over: an antiderivative of z^n is (for ) — the complex case mirrors the real one. Path independence: because z^2 is analytic everywhere, the answer depends only on the endpoints 0 and 1+i , not on the route between them. Whenever the integrand is analytic on a simply connected region, reach for an antiderivative first — it is the fastest tool in complex analysis.

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