4

(a) Suppose $P(x,y)$ and $Q(x,y)$ are conjugate harmonic functions and $C$ is any simple closed curve, prove that $\displaystyle \oint_C P\,dx + Q \,dy = 0$.
(b) If for all simple closed curves $C$ in the region $R$ , $\displaystyle \oint_C P\,dx + Q \,dy = 0$, is it true that $P$ and $Q$ are harmonic conjugate functions? i.e. is converse of (a) true?

For (a) since they are harmonic conjugate functions so they should satisfy Cauchy-Riemann equations and using Green's theorem, it can be shown. The problem is asked under the topic Morera's theorem. Can it be shown using Morera's theorem? How is it related to Morera's theorem?

For (b), the closed loop integral for all curves implies $\displaystyle \frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}$. How show that it is harmonic or non harmonic?

EDIT:: There seems to be a problem which states that if $\displaystyle \oint_C Pdx + Q dy = 0$ for all curves in the region bound by $C$ then show $\displaystyle \oint_C Q\,dx - P \,dy = 0$. I can't prove this either but I think they are equivalent.

Help!!

Community
  • 1
Mula Ko Saag
  • 2,177
  • 1
  • 23
  • 45

1 Answers1

3

(a) When $P$ and $Q$ are conjugate real harmonic functions of $(x,y)$ on a simply connected domain $\Omega\subset{\mathbb C}$ then $f:=P+iQ$ is an analytic function of $z:=x+iy$ on $\Omega$, by definition of "conjugate harmonic". Therefore by Cauchy's theorem for any closed curve $\gamma$ in $\Omega$ one has $$0=\int_\gamma f(z)\ dz=\int_\gamma (P+iQ)\ (dx+i dy)=\int_\gamma (Pdx-Qdy) + i\int_\gamma (Qdx +P dy)\ .$$ It follows that $\int_\gamma (Pdx-Qdy) =0$ for all $\gamma\subset\Omega$. Note the essential hypothesis that $\Omega$ is simply connected. Consider the example $$P(x,y)={y\over x^2+y^2},\quad Q(x,y)={x\over x^2+y^2}\qquad\bigl((x,y)\ne(0,0)\bigr)\ .$$ Then $$P+iQ={ix+y\over x^2+y^2}={i\over z}=:f(z)$$ is analytic on $\Omega:=\dot{\mathbb C}$, but for $\gamma$ the unit circle on has $$\int_\gamma (Pdx-Q dy)={\rm Re}\left(\int_\gamma {i\over z}\ dz\right)=-2\pi\ne0\ .$$ (b) When $\int_\gamma (Pdx+Qdy)=0$ resp. $\int_\gamma (Pdx-Qdy)=0$ for all closed $\gamma\subset\Omega$ this only implies that $(P,Q)=\nabla f$, resp. $(P,-Q)=\nabla f$ for some scalar function $f:\ \Omega\to{\mathbb R}$, but neither $f$ nor its derivatives have to be harmonic.

Christian Blatter
  • 226,825