Summer+2011+examination

1. Evaluate the following integrals:

math \int\limits_{-\infty}^{\infty}\! \frac{\cos(2011x)}{x^2+1} \, d x math

math] [[math ]

Give an example of a M\"{o}bius transformation that maps the upper half-plane math \mathrm{Im} {z}>0 math to the unit disc math math
 * z|<1

Show that every function $f$ analytic on the complex plane and satisfying math \mathrm{Im}\, f(z)>0 math for all z is constant.

Evaluate the following expressions:

\begin{enumerate} \item (5) { $$\int\! \mathrm{Re}\, {z} \, dz$$ over the straight line segment from $z=1+i$ to $z=2+2i$ } \pad \item (5) { $$\int\! e^{\pi z} \, dz$$ over the curve $z(t)=t^2-t + i t$ for $0 \leq t \leq 1$ } \pad \clearpage \item (5) { $$\int\! \tan z \, dz$$ over the circle $|z-i|=3$ } \pad \item (5) { $$\lim\limits_{r\to 0}\int\limits_{C_r}\! \frac{dz}{\sin z}$$ where $C_r$ is the semicircle $z(t)=r e^{it}$ for $0\leq t\leq \pi$ (hint: you can use Laurent series expansion of $1/\sin z$ around $z=0$) } \pad

\end{enumerate}

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\item \begin{enumerate}

\item (5) { Find the number of zeroes of the function $$f(z)=z^6-i z^4 -z^3 -i z^2 - i - 10$$ in the upper half-plane $\mathrm{Im}\, z>0$. } \pad \item (5) { Find the number of zeroes of the function $f(z)=z^{2011}+4z^4-2$ in the disc $|z|<1$. } \pad \clearpage \item (5) { Find the order of the zero of the function $$f(z)=(\cos z - 1)^2(e^{iz}-1)^3\left(\frac{z-\pi}{z}\right)^4$$ at $z=2\pi$ } \pad \item (5) { Give an example of a non-constant analytic function $f(z)$ such that the equation $f(z)=0$ doesn't have any solutions. } \pad \end{enumerate}

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\item { For each of the following statements determine whether it is true or false and write a short explanation supporting your answer (a guess without an explanation will not be graded). \begin{enumerate} \item (4) The function $u(x,y)=e^x \sin y$ is harmonic \pad \item (4) The function $v(x,y)=e^x-e^x\cos y$ is a harmonic conjugate of $u(x,y)=e^x \sin y$ (i.e. $f(x+iy)=u(x,y)+i v(x,y)$ is analytic) \pad \item (3) There exists an analytic (one-valued) function on the complex plane whose square is equal to $e^z-1$

(hint: what could be its order of zero at $z=0$?) \pad \item (3) There exists an analytic (one-valued) function on the complex plane whose square is equal to $\cos^2 z - 1$ \pad \item (3) All values of $(-1)^i$ are real numbers \pad \item (3) There are real numbers among the values of $2^i$ \pad \end{enumerate}