## Complex Velocity Functions

Suppose that is a model flow velocity function with domain The conjugate velocity function is analytic on  may not be simple but if is a simply connected subregion of  has has a primitive satisfying for A function which is a primitive of a complex velocity function is called a complex potential function for the flow. Taking the complex conjugate of (1) above gives for The circulation of along a curve drawn in the fluid is where the integral is evaluated along and the flux of across is where the integral is evaluated along Hence where the integral is evaluated along Provided that %GAMMA lies within the simply connected subregion S on which the complex potential %OMEGA is defined, we may apply the Fundamental Theorem of Algebra to deduce where and are the start and end points of the curve Hence and Hence the circulation or flux along any contour between and depends only on the points %alpha and %beta and is independent of the contour.

If the flow at each point is tangential to a curve then the flow across at each point of is zero. Curves with this property are called streamlines and represent snapshots of the motion of the fluid. If a curve is a streamline then for all points and on so that  