Problem 1674
What are the singularities of \(\displaystyle \pi\ctg\pi z\)? Find the residues at these points. Difficulty: 3. |
Problem 1669
If \(\displaystyle f\in\mathcal{M}(|z|<1)\) then \(\displaystyle f\) has an antiderivative if and only if the residue of \(\displaystyle f\) is \(\displaystyle 0\) at all singularities. Difficulty: 4. |
Problem 1672
\(\displaystyle \frac1{2\pi i}\int_{|z|=2}\tg z\dz=? \) Difficulty: 4. |
Problem 1673
\(\displaystyle \int_\Gamma\frac{\tg z}{z^2+1} \dx =? \) Difficulty: 4. |
Problem 1676
\(\displaystyle \int_\Gamma\frac{\dz}{\cos z}=? \) Difficulty: 4. |
Problem 1677
Let \(\displaystyle \Gamma\) be the curve shown in the figure. (a) Compute \(\displaystyle \displaystyle \int\limits_\Gamma \frac{z^{20}+2}{z^2-1} \dz\). (b) Compute \(\displaystyle \displaystyle \int\limits_{C(0, 1)} \frac {\sin z} z \dz\). Difficulty: 4. |
Problem 1678
\(\displaystyle \int_\Gamma\frac{\dz}{(z-1)^2\sin z}=? \) Difficulty: 4. |
Problem 1681
\(\displaystyle \int_{|z|=2}\dfrac{\sin\frac{\pi}{z}}{z^4-1} = ? \) Difficulty: 4. |
Problem 1682
Let \(\displaystyle 0<r<\pi\). \(\displaystyle \displaystyle\int\limits_{|z|=r}\frac{\dz}{\sin z}=?\) Difficulty: 4. |
Problem 1685
\(\displaystyle \int_{|z-2|=4} \frac{z}{\sin z}\dz = ? \) Difficulty: 4. |
Problem 1670
Calculate the first 6 terms in the Laurent series of \(\displaystyle \ctg z\) and \(\displaystyle \pi\ctg (\pi z)\) on the domain \(\displaystyle 0<|z|<\pi\). What are the residues of \(\displaystyle \displaystyle\frac{\ctg z}z\), \(\displaystyle \displaystyle\frac{\ctg z}{z^2}\), ..., \(\displaystyle \displaystyle\frac{\ctg z}{z^5}\) in \(\displaystyle 0\)? Difficulty: 5. |
Problem 1679
\(\displaystyle \int_\Gamma\frac{\dz}{\bigg(z-\dfrac\pi3\bigg)^2\sin z}=? \) Difficulty: 5. |
Problem 1680
\(\displaystyle \frac1{2\pi i}\int\limits_{|z|=1/4}\frac\dz{\sin\frac1z}=?\) Difficulty: 5. |
Problem 1684
\(\displaystyle \int_{|z|=5}\frac{z^2}{\sin z}\dz=?\) Difficulty: 5. |
Problem 1683
Show that if the complex numbers \(\displaystyle a_1,\ldots,a_n\) are all different and \(\displaystyle p(z)=(z-a_1)\cdot\ldots\cdot(z-a_n)\), then \(\displaystyle \sum_{j=1}^n \frac{p''(a_j)}{(p'(a_j))^3} = 0. \) Difficulty: 7. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |