Path Independence in Complex Line Integrals | Complex Analysis, Part 15

In this episode of the Complex Analysis series, Prof. Happy Strawberry from the F.I.T. Department of Mathematics explores a fundamental question: Does a complex line integral depend on the path? We revisit the integral of f(z)=z from 0 to 1+i, but this time along a different path , first horizontally, then vertically. Step by step, we compute both parts: * First segment: 0 \to 1 * Second segment: 1 \to 1+i Adding the results, we find the same value as before: \int_\gamma z\,dz = i So why didn’t the result change? The answer introduces a key concept: If a function has an antiderivative, the integral depends only on the endpoints — not on the path. This leads to an important condition: * The function must be holomorphic * The domain must be simply connected (no holes) If these conditions fail, path independence breaks and different paths can give different results. In the next episode, we will see exactly what happens when these conditions are violated.