Problem 1562
Does \(\displaystyle e^{-1/z^4}\) have a limit at \(\displaystyle 0\)? Difficulty: 3. |
Problem 1564
Prove that \(\displaystyle \sin (z_1+z_2)=\sin z_1\cos z_2+\cos z_1\sin z_2\) and \(\displaystyle \cos(z_1+z_2)=\cos z_1\cos z_2-\sin z_1\sin z_2.\) Difficulty: 3. |
Problem 1566
Prove that the following equations have only real roots a) \(\displaystyle z\sin z=1\) b) \(\displaystyle \tg z=z\). Difficulty: 3. |
Problem 1560
Show that the only periods of \(\displaystyle \sin z\) are \(\displaystyle 2k\pi\), for \(\displaystyle k\) an integer. Difficulty: 4. |
Problem 1565
Use the Cauchy-product of the series that define the complex exponential to show that \(\displaystyle e^{z+w}=e^ze^w\). Difficulty: 4. |
Problem 1563
Does any of the functions \(\displaystyle e^{iz}\), \(\displaystyle \sin z\), \(\displaystyle \cos z\), \(\displaystyle \tg z\), \(\displaystyle \ctg z\) have a limit as \(\displaystyle \im z\to\pm\infty\)? Difficulty: 5. |
Problem 1561
Let \(\displaystyle D_\varepsilon\) be the domain that one gets by deleting discs with center \(\displaystyle k\pi\) (\(\displaystyle k\in\Z\)) and radius \(\displaystyle \varepsilon<\pi/2\). Show that both \(\displaystyle 1/\sin z\) and \(\displaystyle \ctg z\) are bounded on \(\displaystyle D_\varepsilon\). Difficulty: 6. |
Problem 1559
Let \(\displaystyle f(0)=0\) and \(\displaystyle \displaystyle f(z)=\frac1{\sin z}-\frac1z\) when \(\displaystyle z\ne0\). Is \(\displaystyle f\) differentiable at \(\displaystyle 0\)? Difficulty: 7. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |