Problem 1591
Let \(\displaystyle p(z)=z^n+b_{n-1}z^{n-1}+\dots+b_1z+b_0\) has degree \(\displaystyle n>1\) and no roots in \(\displaystyle |z|>R\). Let \(\displaystyle \displaystyle I(R) = \frac1{2\pi i} \int_{|z|=R}\frac{dz}{p(z)}\). Show that (a) \(\displaystyle \displaystyle\lim_{R\to\infty}I(R)=0\); (b) \(\displaystyle I(R)\) is constant. (c) \(\displaystyle I(R)=0\). Difficulty: 5. |
Problem 1592
Find the following integrals! a) \(\displaystyle \displaystyle\int\limits_{[0, 1+i]} e^z dz\) b) \(\displaystyle \displaystyle\int\limits_{|z|=1} \frac{1}{z}dz\) c) \(\displaystyle \displaystyle\int\limits_{|z|=2} \frac{\dz}{z^2 +1}\) (\(\displaystyle T\) is the square with vertices \(\displaystyle \pm1\pm i\) oriented positively.) Difficulty: 5. |
Problem 1595
Let \(\displaystyle a\) and \(\displaystyle b\) be different complex numbers. Show that on \(\displaystyle \C \setminus[a,b]\) there is a holomorphic branch of \(\displaystyle \log\frac{z-a}{z-b}\). Difficulty: 5. |
Problem 1593
Let \(\displaystyle D\) be a simply connected domain that does not contain the origin. (a) Show that \(\displaystyle 1/z\) has an antiderivative on \(\displaystyle D\). (b) Show that if \(\displaystyle g'(z)=1/z\) on \(\displaystyle D\), then \(\displaystyle ze^{-g(z)}\) is constant. (c) Show that \(\displaystyle \log z\) has a continuous branch on \(\displaystyle D\). Difficulty: 6. |
Problem 1594
Let \(\displaystyle D\) be a simply connected domain and \(\displaystyle f(z)\) a non-vanishing holomorphic function on \(\displaystyle D\). (a) Show that \(\displaystyle f'(z)/f(z)\) has an antiderivative on \(\displaystyle D\). (b) Show that if \(\displaystyle g'=f'/f\) on \(\displaystyle D\), then \(\displaystyle f(z)e^{-g(z)}\) is constant on \(\displaystyle D\). (c) Show that \(\displaystyle \log f\) has a continuous branch on \(\displaystyle D\). Difficulty: 6. |
Problem 1590
Show that for all \(\displaystyle a \in \C\) \(\displaystyle \int_{-\infty}^\infty e^{-x^2/2}\cdot e^{iax}\dx = \sqrt{2\pi}\cdot e^{-a^2/2}. \) Difficulty: 7. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |