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\noindent{\it Math 6720}
\hfill{\bf Appl. Complex Var., Asymptc Mthds}\hfill
{\it Alexander Balk}


\bigskip\bigskip

\noindent{\bf HW6}\hfill{\bf\large NAME:
.....................................................................}
\hfill {\bf due Wednesday, 2/21/2018}
\begin{enumerate}

\item {\bf Jordan's lemma} to neglect integrals
\begin{eqnarray*}
  I(R)=\int_{C_R}f(z)\; e^{i a z} \; dz\quad\quad (a>0)
\quad\mbox{over ``large'' semi-circle $C_R: \; z=Re^{i\phi}\;\;(0\le\phi\le\pi)$, $\quad R\rightarrow\infty$.}
\end{eqnarray*}

{\bf Prove:}\hspace*{1.3cm}
If on $C_R$, $\quad|f(z)|<F(R)\rightarrow0\quad$  
[for some function $F(R)$], 
\hspace*{1cm} then $I(R)\rightarrow 0$. 



\newpage\item The (single-valued) function $f(z)=\sqrt{z(z-1)(z-4)}$ is defined by\\
branch cuts \quad $z=-i t,\quad z=1+i t, \quad z=4-i t\quad (0\le t <\infty)$\quad
and value\quad $f(2)=-2i$.

\bigskip

Evaluate
\begin{enumerate}
\item$f(-3)$,
\item$f(1/2)$,
\item$f(5)$.
\end{enumerate}

\newpage\item 
The (single-valued) function 
\begin{eqnarray*}
f(z)=\frac{(z-16)^{1/3}(z-20)^{1/2}}{z-1}
\end{eqnarray*}
is defined by branch cuts \quad $z=16+i t$,\quad $z=20-i t$ \quad $(0\le t <\infty)$
and value $f(24)=\frac{4}{23}$.

Evaluate $\oint_C f(z) dz$ over the circle $C: \; |z-2|=2$.

\newpage\item 
We are given multivalued analytic function
 $$f(z)=[(z-1)(z-2)]^{1/3}.$$
What branch cuts would you make to separate single-valued branches?
%Would the cut from $z=1$ to $z=2$ along the real axis work?

%\newpage\item
%Let a function $f(z)$ be some branch of $z^{1/n}$ ($n$ is an integer). 
%Show that $f'(z)=\frac{f(z)}{n\; z}$.\\
%In other words, the function and its derivative are determined by {\it the same branch} of $z^{1/n}$.

%\newpage\item
%The (single-valued) function $f(z)=\sqrt{z(z-1)}$ is defined by
%the branch cut $z=t\;(0\le t\le 1)$ and the condition that 
%$f$ is positive for real $z>1$.

%\begin{enumerate}
%\item Find the first three non-zero terms of its Laurent series (centered
%at $z_0=0$) in the domain $|z|>1$.
%\item Where does the series converge?
%\end{enumerate}

\newpage\item{\bf Describe singularities (their locations, types, and orders).}  
\begin{enumerate}
\item Describe singularities (their locations and types) for the
  function $ f(z)=\frac{(z-1)^2\;\cos^2 (z/2)}{(z+1)^3\; z(z-\pi)}.$

\item Consider the multi-valued function $f(z)=\frac{\sqrt{z}-1}{z-1}.$
To separate $f(z)$ into single-valued branches, make a brunch cut along the negative real semi-axis. 
What isolated singularities (their locations and types) do these branches have?

\item Consider the multi-valued function $f(z)=(1+z)^{1/z}.$ 
To separate $f(z)$ into single-valued branches, make a brunch cut $z=-t\; (1\le t<\infty)$. 
What isolated singularities (their locations and types) do these branches have?

\end{enumerate}
\end{enumerate}

\end{document}

\newpage\item
The function $f(z)$ is analytic in the entire complex plane except at
$z=i/2$, where it has a simple pole, and at $z=2$, where it has a
double pole. It is known that
\begin{eqnarray*}
\oint_{|z|=1}f(z)dz &=& 2\pi i\,,\\
\oint_{|z|=3}f(z)dz &=& 0\,, \\
\oint_{|z|=3}f(z)(z-2)dz &=& 0\,,
\end{eqnarray*}
and $f(z)$ is bounded at infinity 
(i.e. $\exists M>0, \;\exists R>0: |z|>R\;\Rightarrow\; |f(z)|<M$).

\medskip

Find $f(z)$ (unique up to an arbitrary additive constant).

{\footnotesize{\it Suggestion:} 
Find an examle $f_0(z)$ of such function and use the Liouville theorem
to show that $f(z)-f_0(z)$ is constant.}

\newpage\item 
Prove that if all singularities of an analytic function $f(z)$ 
in the extended complex plane are poles, then $f(z)$ is a rational function 
(i.e. the ratio of two polynomials).

{\footnotesize{\it Suggestion:} 
First show that $f(z)$ can have only finite number of poles. Then for each
pole at finite $z$ take the negative power part of the corresponding Laurent
expansion; for $z=\infty$, take the positive power part; add all of
them to get a rational function $R(z)$. Finally, apply the Liouville theorem
to show that $f(z)-R(z)$ is a constant.}


\end{enumerate}



\end{document}



\newpage\item
Evaluate the integral
\begin{eqnarray*}
\int_{C}z^{-1/3}dz
\end{eqnarray*}
over the path $C$ traversing from $z=1+i$ to $z=1-i$ along the
following composite curve $C=C_1+C_2+C_3$
\begin{eqnarray*}
&&C_1:\quad \mbox{hyperbola } 2y^2-x^2=1 
          \mbox{ from } z=1+i \mbox{ to } z=-2+i\sqrt{5/2},\\
&&C_2:\quad \mbox{segment } x=-2 
          \mbox{ from } z=-2+i\sqrt{5/2} \mbox{ to } z=-2-i\sqrt{5/2},\\
&&C_3:\quad \mbox{hyperbola } 2y^2-x^2=1 
          \mbox{ from } z=-2-i\sqrt{5/2} \mbox{ to } z=1-i.
\end{eqnarray*}

Here $z^{-1/3}$ denotes the branch defined by the branch cut along the positive $x$-axis 
and the condition that it is real when $z$ is negative.



Suppose we would like to approximate the branches of the multi-valued function 
$$f(z)=(1+z)^{1/z}$$ for small $z\rightarrow 0$ by three-term expansions
 (i.e. the ones with 3 non-zero terms) in powers of $z$ . 
Find the three-term approximation for that branch 
of this multi-valued function which takes real values for all positive $z$.
 
Is it possible to find the three-term expansions approximating the
 other branches?





\newpage\item 
\item {\bf Describe singularities (their locations, types, and orders).}  
\begin{enumerate}
\item What are the singularities of $f(z)=\frac{[(z+1)^{10}-1]\;\cos^2 (z/2)}{(z+1)^3\; z\; (z-\pi)^3}.$
\item Consider the multi-valued function $f(z)=\frac{\sqrt{z}-1}{z-1}.$
To separate $f(z)$ into single-valued branches, make a brunch cut along the negative real semi-axis. 
What {\it isolated} singularities do these branches have?
\item Consider the multi-valued function $f(z)=(1+z)^{1/z}.$
To separate $f(z)$ into single-valued branches,  make a brunch cut along the negative real semi-axis. 
What {\it isolated} singularities do these branches have?
\end{enumerate}

\newpage\item {\bf The singularities of Euler's gamma-function $\Gamma(z)$.}
\begin{enumerate}
\item What are the singularities of $\Gamma(z)$?
\item What are the residues of this function (at all singularities)?
\end{enumerate}


\newpage\item
The function $f(z)$ is analytic in the entire complex plane except at
$z=i/2$, where it has a simple pole, and at $z=2$, where it has a
double pole. It is known that
\begin{eqnarray*}
\oint_{|z|=1}f(z)dz &=& 2\pi i\,,\\
\oint_{|z|=3}f(z)dz &=& 0\,, \\
\oint_{|z|=3}f(z)(z-2)dz &=& 0\,,
\end{eqnarray*}
and $f(z)$ is bounded at infinity 
(i.e. $\exists M>0, \;\exists R>0: |z|>R\;\Rightarrow\; |f(z)|<M$).

\medskip

Find $f(z)$ (unique up to an arbitrary additive constant).

{\footnotesize{\it Suggestion:} 
Find an examle $f_0(z)$ of such function and use the Liouville theorem
to show that $f(z)-f_0(z)$ is a constant.}

\newpage\item 
Prove that if all singularities of an analytic function $f(z)$ 
in the extended complex plane $\overline{\mathbb{C}}$ are poles, then $f(z)$ is a rational function 
(i.e. the ratio of two polynomials).

{\footnotesize{\it Suggestion:} 
First show that $f(z)$ can have only finite number of poles. Then for each
pole at finite $z_0$ take the negative power part of the corresponding Laurent
expansion; for $z_0=\infty$, take the positive power part; add all of
them to get a rational function $R(z)$. Finally, apply the Liouville theorem
to show that $f(z)-R(z)$ is a constant.}

\newpage\item Evaluate the integrals
\begin{eqnarray*}
  \int_{-\infty}^{\infty}\frac{\cos k x }{(x+b)^2+a^2} \; dx= 
\frac{\pi}{a}\; e^{-k a}\; \cos b k\;,\\
  \int_{-\infty}^{\infty}\frac{\sin k x }{(x+b)^2+a^2} \; dx= 
-\frac{\pi}{a}\; e^{-k a}\; \sin b k\; 
\end{eqnarray*}
(with parameters $k>0,\, a>0,\; b$ real).

\newpage\item Integrate
\begin{eqnarray*}
 \int_{-\infty}^{\infty}\frac{\cos a x \;-\; \cos b x}{x^2}\;dx\;\;=\;\; \pi(b-a)
\end{eqnarray*}
(with parameters $a\ge0,\; b\ge0$).

\newpage\item
Integrate
\begin{eqnarray*}
\int_{-\infty}^{\infty}\frac{\sin\alpha x}{\sinh\pi x}\;dx\;\;=
\;\; \tanh\frac{\alpha}{2}
\end{eqnarray*}
In which strip $a<\mbox{Re}(\alpha)<b$ of complex parameter $\alpha$ does this formula hold?

{\footnotesize{\it Suggestion:} Notice that $\sinh\pi (z+i)=-\sinh\pi z$ 
and use a long rectangular contour with indentations. 
Replace $\sin\alpha x$ by the exponential function (at the end you will take the imaginary part)
and notice that when $x$ is replaced by $x+i$ the new integrand is multiplied by a constant factor, 
independent of $x$.\\
At first, assume real $\alpha$; 
then use analytic continuation to establish this formula for complex $\alpha$.}

\end{enumerate}

\end{document}



\item {\bf Expansion in partial fraction.} 
Show that if $a,\; b,\; c$ are distinct complex numbers, and
  $P(z)$ is a polynomial of degree at most 3, then
\begin{eqnarray*}
  f(z)\stackrel{\rm def}{=} \frac{P(z)}{(z-a)(z-b)(z-c)}\;\;=\;\;
\frac{A}{z-a}+\frac{B}{z-b}+\frac{C}{z-c}+E
\end{eqnarray*}
with constants
\begin{eqnarray*}
A&=&\mbox{Res}f(a)=\frac{P(a)}{(a-b)(a-c)},\\
B&=&\mbox{Res}f(b)=\frac{P(b)}{(b-a)(b-c)},\\
C&=&\mbox{Res}f(c)=\frac{P(c)}{(c-a)(c-b)},\\\\
E&=&\lim_{z\rightarrow\infty}f(z)
\end{eqnarray*}

 
\end{enumerate}

\end{document}


