You can visualize the "frantic activity" or flow around poles (singularities) of a function. For example, a simple pole acts like a source or sink for fluid in the field. Visual Analysis: It is a central tool in Visual Complex Analysis
If residue ( = a+ib ), then circulation ( = -2\pi b ), flux ( = 2\pi a ). A purely real residue (b=0) gives flux only—a source. A purely imaginary residue (a=0) gives circulation only—a vortex. polya vector field
Understanding the Pólya Vector Field: A Bridge Between Complex Analysis and Fluid Dynamics You can visualize the "frantic activity" or flow
Hence:
Indeed, the stream function (\psi) such that (\mathbfV_f = ( \psi_y, -\psi_x )) can be taken as (\psi = -v). Check: [ \psi_y = -v_y = -(-u_x) = u_x? \text Wait carefully. ] Better: Let (\psi = -v). Then (\nabla^\perp \psi = (\psi_y, -\psi_x) = (-v_y, v_x)). But by Cauchy–Riemann, (v_x = u_y), (v_y = -u_x), so ((-v_y, v_x) = (u_x, u_y)) — that’s (\nabla u), not (\mathbfV_f). So that’s not correct. Let's derive cleanly: A purely real residue (b=0) gives flux only—a source