Problem 1606
\(\displaystyle \int\limits_{|z|=2} \frac{3^z}{(z-1)^2(z+3)^2}\dz =? \) Difficulty: 2. |
Problem 1607
The function \(\displaystyle f(z)\) is holomorphic in the interior of the unit disc (\(\displaystyle |z|<1\)) and \(\displaystyle |f|<1\). How large can \(\displaystyle |f'''(0)|\) be? Difficulty: 2. Answer (final result) is provided for this problem. |
Problem 1609
\(\displaystyle \frac1{2\pi i}\int_{|z|=5}\frac{\cos z}{z^2}\dz = ? \qquad \int_{|z|=3}\frac{e^z}{z^8}\dz = ? \qquad \int_{|z|=3}\frac{e^z}{(z-2)^3}\dz = ? \) Difficulty: 3. |
Problem 1610
For \(\displaystyle a,r>0\) find the following integrals: \(\displaystyle \frac1{2\pi i}\int\limits_{|z|=r}a^z\dz; \quad \frac1{2\pi i}\int\limits_{|z|=r}\frac{a^z}{z}\dz; \quad \frac1{2\pi i}\int\limits_{|z|=r}\frac{a^z}{z+1}\dz; \quad \frac1{2\pi i}\int\limits_{|z|=r}\frac{a^z}{z^2}\dz; \quad \frac1{2\pi i}\int\limits_{|z|=r}\frac{a^z}{(z+2)^2}\dz. \) Difficulty: 3. |
Problem 1604
\(\displaystyle \frac1{2\pi i}\int_{|z|=5}\frac{\cos z}z\dz = ? \qquad \int_{|z|=3}\frac{e^z}z\dz = ? \qquad \int_{|z|=3}\frac{e^z}{z-2}\dz = ? \qquad \int_{|z|=3}\frac{e^z}{(z-2)(z-4)}\dz = ? \) Difficulty: 4. |
Problem 1608
Show that if \(\displaystyle f\in O(|z|\le1)\), then a) \(\displaystyle f'(z)\left(1-|z|\right)\) is bounded b) What can we say about the \(\displaystyle n\)-th derivative? Difficulty: 5. |
Problem 1601
Let \(\displaystyle f\) be continuous on the closed unit disc and holomorphic in its interior. Prove that for \(\displaystyle |z|<1\) \(\displaystyle f(z)=\frac1{2\pi i}\int_{|z|=1}\frac{f(\xi)}{z-\xi}\dxi. \) Difficulty: 6. |
Problem 1603
Let \(\displaystyle n \in \Z\). Find \(\displaystyle \int_{|z|=2}\frac{z^n}{(z-1)(z-3)}\dz . \) Difficulty: 7. |
Problem 1605
Let \(\displaystyle a,b \in \C\) and \(\displaystyle |b|<1\). Prove that \(\displaystyle \frac1{2\pi}\int_{|z|=1}\left|\frac{z-a}{z-b}\right|^2|\dz| = \frac{|a-b|^2}{1-|b|^2}+1. \) Difficulty: 7. Solution is available for this problem. |
Problem 1599
Let \(\displaystyle f\) be a holomorphic function on the disc \(\displaystyle |z|<1+\varepsilon\) and let \(\displaystyle |a|<1\). Find a function \(\displaystyle \varphi_a:[0,2\pi]\to\RR\) such that \(\displaystyle f(a)=\frac1{2\pi}\int_0^{2\pi}f(e^{it})\varphi_a(t)dt.\) Difficulty: 8. |
Problem 1600
Prove for any complex number \(\displaystyle a\) that \(\displaystyle \frac1{2\pi} \int_0^{2\pi} \log\big|e^{it}+a\big| \dt = \begin{cases} \log|a| & \text{if $|a|>1$,} \\ 0 & \text{if $|a|\le1$.} \\ \end{cases} \) Difficulty: 8. |
Problem 1602
Let \(\displaystyle f\) be a holomorphic function on the disc \(\displaystyle |z|<1+\varepsilon\). Prove that \(\displaystyle \log |f(0)|\le \frac1{2\pi}\int_0^{2\pi}\log|f(e^{it})|\dt.\) When does equality hold? Difficulty: 8. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |