Problem 1735
How many zeros does the function \(\displaystyle \cos z=2z^3\) have in the unit disc? Difficulty: 3. |
Problem 1737
How many zeros do the functions have on the given domain? (a) \(\displaystyle \sin z=2z^2\), \(\displaystyle |z|<1\) (b) \(\displaystyle z^4+z^3-4z+1=0\), \(\displaystyle 1<|z|<2\) (c) \(\displaystyle z^6-6z+10\), \(\displaystyle |z|>1\). Difficulty: 3. |
Problem 1738
Let \(\displaystyle |a|=3\). Find the number of zeros of \(\displaystyle z^4+z^3+az-1\) in the domain \(\displaystyle 1<|z|<2\)? Difficulty: 3. |
Problem 1740
How many zeros does \(\displaystyle 2^z+3z^2-z\) have in the unit disc? Difficulty: 3. |
Problem 1743
Prove that \(\displaystyle az^n+3z+1\) has a root in the unit disc for any \(\displaystyle a\in \C\). Difficulty: 4. |
Problem 1741
Prove the fundamental theorem of algebra from Rouche's theorem. Difficulty: 5. |
Problem 1744
Let \(\displaystyle a\in\CC\), \(\displaystyle |a|<1\), \(\displaystyle n\in\NN\). Show that \(\displaystyle (z-1)^ne^z=a\) has exactly \(\displaystyle n\) solutions in the half-plane \(\displaystyle \re z>0\). Difficulty: 5. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |