Antiderivatives in ℂ

Complex Analysis · Axiom Academy

LESSON Antiderivatives in the Complex Plane When does a complex function have an antiderivative — and what does it buy you? It all comes down to whether the contour integral cares about the path. 1. An Antiderivative Makes the Path Irrelevant Suppose f has an antiderivative F (so F'=f ) on a domain containing the contour . Then the contour integral collapses to the same subtraction as in real calculus — it depends only on the endpoints z_0 and z_1 , not on how gets between them. Fundamental Theorem for contour integrals — endpoints only Integrate from z_0 = 0 to z_1 = 1+i . The theorem gives — and you get exactly i whether you go straight, take the L-shaped detour, or any curve between the two points. 2. When Does an Antiderivative Exist? Path-independence has a clean test. Glue any two paths from z_0 to z_1 into a single closed loop (out along one, back along the other). The two integrals agree exactly when the loop integral is zero — so an antiderivative exists precisely when every closed loop integrates to 0 . You can undo the derivative — there is an analytic F with F' = f on the whole domain. Any two contours sharing endpoints give the same value. for each closed contour C in the domain. If f is analytic on a simply connected domain (no holes), all three hold automatically — an antiderivative is guaranteed. 3. Where It Breaks: f(z) = 1/z

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