Problem 1688
Find the singularities and residues of the following functions: \(\displaystyle \frac1z; \quad \frac1{z^2}; \quad \frac1{z^2+2z}; \quad \quad \frac1{\sin z}; \quad \sin\frac1z; \quad \frac{e^z}{z^2+4}; \quad \frac{e^z}{(z^2+4)^2}; \quad \frac{e^z}{(z^2+4)^3} \quad \frac{e^z-z^3+8}{z^2+1} \) Difficulty: 4. |
Problem 1692
\(\displaystyle \frac1{2\pi i}\int_{|z|=5}\tg z\dz=? \) Difficulty: 4. |
Problem 1686
Find the residues of \(\displaystyle \tg z\), \(\displaystyle \tg^2z\), \(\displaystyle \tg^3z\) in \(\displaystyle \frac{3\pi}2\). Difficulty: 5. |
Problem 1687
What are the residues of \(\displaystyle \displaystyle\frac{\tg z}{1-\cos z}\) and \(\displaystyle \displaystyle\frac{e^z}{\tg z-\sin z}\) in \(\displaystyle 0\)? Difficulty: 5. |
Problem 1689
Let \(\displaystyle f\) and \(\displaystyle g\) be holomorphic in a neighborhood of \(\displaystyle z_0\). (a) Assume that \(\displaystyle g\) has a simple zero in \(\displaystyle z_0\). Prove that \(\displaystyle \displaystyle\res_{z_0}\frac{f}{g}=\frac{f(z_0)}{g'(z_0)}\). (b) Assume that \(\displaystyle g\) has a double zero in \(\displaystyle z_0\). Express \(\displaystyle \displaystyle\res_{z_0}\frac{f}{g}\) in terms of Taylor coefficients of \(\displaystyle f\) and \(\displaystyle g\). Difficulty: 5. |
Problem 1690
\(\displaystyle \int_\Gamma\frac{\ctg z}{z^8-z^6-z^4+z^2}\dz=? \) Difficulty: 7. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |