# Summer 2011 examination

1.
Evaluate the following integrals:

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

[[math]$] Give an example of a M\"{o}bius transformation that maps the upper half-plane$\mathrm{Im} {z}>0$to the unit disc$|z|<1$Show that every function f analytic on the complex plane and satisfying$\mathrm{Im}\, f(z)>0math
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 $z=2+2i$
}
\item (5)
{
$$\int\! e^{\pi z} \, dz$$ over the curve $z(t)=t^2-t + i t$ for $0 \leq t \leq 1$
}
\clearpage
\item (5)
{
$$\int\! \tan z \, dz$$ over the circle $|z-i|=3$
}
\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 $z=0$)
}

\end{enumerate}

\nextpage

\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 $\mathrm{Im}\, z>0$.
}
\item (5)
{
Find the number of zeroes of the function $f(z)=z^{2011}+4z^4-2$ in the disc $|z|<1$.
}
\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 $z=2\pi$
}
\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.
}
\end{enumerate}

\nextpage

\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
\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)
\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 $z=0$?)
\item (3) There exists an analytic (one-valued) function on the complex plane whose square is equal to $\cos^2 z - 1$
\item (3) All values of $(-1)^i$ are real numbers
\item (3) There are real numbers among the values of $2^i$