# proof of closed curve theorem

Let

 $f(x+iy)=u(x,y)+iv(x,y).$

Hence we have

 $\int_{C}f(z)\,dz=\int_{C}\omega+i\int_{C}\eta$

where $\omega$ and $\eta$ are the differential forms

 $\omega=u(x,y)\,dx-v(x,y)\,dy,\qquad\eta=v(x,y)\,dx+u(x,y)\,dy.$

Notice that by Cauchy-Riemann equations $\omega$ and $\eta$ are closed differential forms. Hence by the lemma on closed differential forms on a simply connected domain we get

 $\int_{C_{1}}\omega=\int_{C_{2}}\omega,\quad\int_{C_{1}}\eta=\int_{C_{2}}\eta.$

and hence

 $\int_{C_{1}}f(z)\,dz=\int_{C_{2}}f(z)\,dz$
Title proof of closed curve theorem ProofOfClosedCurveTheorem 2013-03-22 13:33:34 2013-03-22 13:33:34 paolini (1187) paolini (1187) 6 paolini (1187) Proof msc 30E20