Problem 1618
Find the Taylor series of \(\displaystyle \displaystyle\frac{z^2+i}{z^2+z}\) around \(\displaystyle i\). Difficulty: 3. |
Problem 1617
Let \(\displaystyle f\in O(\CC)\). Then \(\displaystyle \re f\) can not be bounded either form below or above. Difficulty: 4. |
Problem 1620
Describe those \(\displaystyle f\in O(\CC)\) which do not take positive values. Difficulty: 4. |
Problem 1619
Find the Taylor series of \(\displaystyle (1+x)^c=\exp\big(c\cdot\log(1+z)\big)\) around \(\displaystyle 0\). Difficulty: 5. |
Problem 1622
Assume that \(\displaystyle f:\CC\leftrightarrow \CC\) is a biholomorphism. Show that \(\displaystyle f(z)=Az+B\). Difficulty: 6. |
Problem 1615
Show that if \(\displaystyle f\) is a double peridodic entire function (i.e. \(\displaystyle f(z+a)=f(z), f(z+b)=f(z)\) where \(\displaystyle a\) and \(\displaystyle b\) are linearly independent over \(\displaystyle \Q\), then \(\displaystyle f\) is constant. Difficulty: 7. |
Problem 1614
Prove that if \(\displaystyle f\) is entire and its image is disjoint from the real interval \(\displaystyle [-1\), \(\displaystyle 1]\), then \(\displaystyle f\) is constant. Difficulty: 9. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |