Problem 1543
Show that if \(\displaystyle f(z)\) is differentiable at \(\displaystyle z_0\), then so is \(\displaystyle g(z):=\overline{f(\overline{z}})\) at \(\displaystyle \overline{z_0}\). Difficulty: 4. |
Problem 1544
If \(\displaystyle f\) is entire, then so is \(\displaystyle g(z):=\overline{f(\bar{z}})\). Difficulty: 4. |
Problem 1545
Let \(\displaystyle D\subset\RR^2\) be an open domain and \(\displaystyle u,v:D\to\RR^2\) twice differentiable for which the map \(\displaystyle x+yi\mapsto u(x,y)+iv(x,y)\) is regular on \(\displaystyle D\). Show that \(\displaystyle \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0. \) Difficulty: 5. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |