Complex Analysis: Theory and Applications · Problem 3.2.6

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · irrotational-solenoidal-vector-fields

Problem

A vector field F(z)=U(x,y)+iV(x,y)\mathbf{F}(z) = U(x,y) + iV(x,y) is said to be irrotational if Uy(x,y)=Vx(x,y)U_y(x,y) = V_x(x,y). It is said to be solenoidal if Ux(x,y)=−Vy(x,y)U_x(x,y) = -V_y(x,y). If f(z)f(z) is an analytic function, show that F(z)=f(z)‾\mathbf{F}(z) = \overline{f(z)} is both irrotational and solenoidal.

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