Contours and Curves

Complex Analysis · Axiom Academy

Before you can integrate in the complex plane, you need the path — and a contour is just a point in motion, traced by a single function z(t) . 1. A Path Is a Point in Motion A curve in the complex plane is a continuous function . Feed it the parameter t and out comes a complex number — a position. As t runs from a to b , that point sweeps out the path, from the initial point to the terminal point . The unit circle, traced once counterclockwise Start and end coincide — a closed contour 2. The Derivative Is the Direction of Travel Differentiate the parametrization and you get the tangent vector — the velocity of the moving point. It is glued to the path, pointing the way t increases. A curve is smooth when is continuous and never zero, so this arrow always has a clear direction. points along the path the way the parameter advances — the orientation. For , the velocity is — always perpendicular to the radius. everywhere, so the circle is swept at unit speed. Run it as , i.e. , and every tangent flips — the arrow turns around. The segment from 0 to 1+i is , . Its tangent is the constant — the direction never changes, exactly as you'd expect for a straight line. 3. Integrating Along the Contour Now the two ingredients combine. To integrate f along a contour, substitute the position and the velocity , turning a complex contour integral into an ordinary integral in the real variable t : The arc length is the same idea with f=1 : . For the unit circle, — its circumference.

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