Let \(f\) be holomorphic on a neighborhood of the closed unit disk and write \[ u(x,y)=\operatorname{Re}f(x+iy). \] Prove that \[ \int_C u_y\,dx-u_x\,dy=0, \] where \(C\) is the unit circle.
Solution
Write \[ f(x+iy)=u(x,y)+iv(x,y). \]
1. On a neighborhood of the closed unit disk,\[ u_y\,dx-u_x\,dy=-dv. \]
Since \(f\) is holomorphic, the Cauchy–Riemann equations give \[ u_x=v_y, \qquad u_y=-v_x. \] Therefore \[ u_y\,dx-u_x\,dy =-v_x\,dx-v_y\,dy =-dv. \]
2. \(\displaystyle\int_C u_y\,dx-u_x\,dy=0\).
Let \(\gamma:[0,2\pi]\to C\) be any smooth parametrization with \(\gamma(0)=\gamma(2\pi)\). By step 1, \[ \int_C u_y\,dx-u_x\,dy =-\int_0^{2\pi}\frac{d}{dt}(v\circ\gamma)(t)\,dt =-v(\gamma(2\pi))+v(\gamma(0)) =0. \]
3. Q.E.D.
Step 2 is the required identity.