Hi all there, Would you please help me with these questions please. How to prove: (1) that if f is a holomorphic function in a domain D(in the complex plane), then −log |f| is lower-semicontinuous? (Deinition of lower-semicontinuity: A function \( u : K\subset\mathbb{C}\rightarrow\mathbb{R} \cup \{+\infty\}\) is called lower semi-continuous (l.s.c.) at a point \( z \in \mathbb{C} \) if \(\liminf_{t\rightarrow z, t\in K} u(t)\ge u(z) \) (2) that log |f| is harmonic (i.e. \(\Delta (\log |f|)=0\)) where \(|f|\ne 0.\) (3) If f is holomorphic on D, is it true that Re(f) (i.e., real part of f) is C-infinity (i.e., infinitely differentiable)? Thanks [/tex]
Hi soma, I would be happy to help you out, but you have to make an effort first. Here are some pointers to get you started. 1) Holomorphic functions are very nice functions. What properties of such functions would make you think the proposition is true or false? 2) Just calculate what the Laplacian of log|f| is. 3) What do you think? Again, holomorphic functions are very nice, so what properties of these nice functions might you use to prove the statement. Alternatively, is there any reason to think this statement wouldn't be true. P.S. Unfortunately, there is no LaTeX on this forum.
hi, thanks for ur reply. I understand these facts: 1) if |f| not equal to zero, log |f| is continuous---> lower semicontinuos. what if |f| =0? 2) How do I calculate the Laplacian for an arbitrary function? Please Register or Log in to view the hidden image! 3) If f=u+ive, then Re(f)=u. This u must be infinitely differentiable as f being holomorphic is infinitely differentiable. I hope this is right, isn't it? Thanks
Hi soma, 1) Good, - log |f| is continuous so as long as f is not 0. For f = 0, just apply the definition directly. It should be intuitively clear that - log |f| is "continuous" at f = 0 i.e. it doesn't wiggle around or do anything except go to infinity in a nice regular way. 2) What do you mean? To calculate the Laplacian you just start taking partial derivatives! Please Register or Log in to view the hidden image! Now what I think you want to ask is how the general mess you will obtain will tell you anything? Obviously it isn't going to true for just any old u and v, so you're going to need to use the fact that u and v come from a holomorphic function. The Cauchy-Riemann equations will undoubtedly prove useful. 3) Good, holomorphic functions are infinitely differentiable on their domain. Can you think of a way to explicitly represent various partial derivatives of u in terms of derivatives of f?
