I don't understand a statement of Ahlfors's Complex Analysis on page 163 in Harmonic Functions:

If $u$ is a harmonic function and $\gamma$ is a regular curve with equation $z=z(t)$ and $\frac{\partial}{\partial n}$ is the normal that points to the right of the tangent, then we obtain $\ast du = \frac{\partial u}{\partial n} |dz|$. Here, $\ast du$ means $-\frac{\partial u}{\partial y} dx + \frac{\partial u}{\partial x}dy$.

I don't understand this statement at all because the left hand side and right hand side don't seem to be the same object at all. $\ast du$ is a one-form, but $\frac{\partial u}{\partial n} |dz|$ seems to be a number $\frac{\partial u}{\partial n}$ times a metric $|dz|$. Right-hand side doesn't look like a 1-form to me. Can anyone explain what Ahlfors means?


Of course they are different things. $\ast du=-u_ydx+u_xdy$ is defined on $\Omega$ whenever $u_x, u_y$ exist. While on the other hand $\frac{\partial u}{\partial n} |dz|$ is not defined if a regular curve $\gamma $ is not given. When $\gamma $ is not given, "normal derivative with respect to $\gamma$" has no meanings.

But once $\gamma $ is given, $\frac{\partial u}{\partial n} |dz|$ can be defined on $\gamma $ and

\begin{align} &\frac{\partial u}{\partial n}=u_x\cos \beta +u_y\sin\beta =u_x\sin\alpha -u_y\cos \alpha ,\\ &\frac{\partial u}{\partial n} |dz|=u_x|dz|\sin \alpha -u_y|dz|\cos \alpha =u_xdy-u_ydx, \end{align} since $|dz|\cos \alpha =dx,$ $ |dz|\sin \alpha =dy$. ($\alpha , \beta $ are the same as in Ahlfors p.163.)

Therefore $$ \ast du = \frac{\partial u}{\partial n} |dz| $$ holds on $\gamma $ and $$ \int_\gamma \ast du = \int_\gamma \frac{\partial u}{\partial n} |dz| $$ holds for every regular curve $\gamma$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.