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\begin{center}
{\huge  \sc  Math 122A Homework 3 } \\
{Due date: 1/28/21}\\
{Name: }\\
{\large  }
\end{center}

\begin{enumerate}
\item[3.6.6] Find the radius of convergence of the following series.
 \begin{enumerate}[(i)]
       
        
        \item $\sum z^n/n$
        
        \item $\sum z^n/n!$
        
        \item $\sum n!z^n$
        
        \item $\sum z^nn^k$ where $k$ is a positive integer.
        
        \item$ \sum z^{n!}$
        
        
    \end{enumerate}

\newpage

    \item [3.6.8] Find the radius of convergence of the following series:
    
   \begin{enumerate}[(i)]
       
        
        \item $z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+ \ldots  $
        
         \item $1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+ \ldots  $
         
          \item $z-\frac{z^2}{2!}+\frac{z^3}{3!}-\frac{z^4}{4!}+ \ldots  $
          
          \item $1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+ \ldots  $
          
          \item $ 1+az +\frac{a(a-1)}{2!}z^2 + \ldots + \frac{a(a-1)\cdots (a-n+1)}{n!}z^n + \ldots$
          
         (Note that in part (iv) the radius of convergence may differ for different values of $a$. )
        
    \end{enumerate}

    
    \newpage
    
    \item [3.6.12]  Prove that if each of the series $\sum a_nz^n,\sum b_nz^n $ and $\sum a_nb_nz^n$ has radius of convergence equal to 1, then the series $\sum a_nb^2_nz^n$ and $\sum a^2_nb_nz^n$.
    
    \newpage
    
    \item [3.6.15] Suppose that the power series $\sum_{n=0}^{\infty}a_nz^n$ has a recurring sequence of coefficients: that is, $a_{n+k}=a_n$ for some fixed positive integer $k$ and all $n$. Prove that the series converges for $|z|<1$ to a rational function $p(z)/q(z)$ where $p,q$ are polynomials, and that the roots of $q$ are all on the unit circle. What happens if $a_{n+k}=a_n/k$?
    
    \newpage
    
    \item [4.7.2]
    
    Show that $f(z)=|z|$ is continuous everywhere and differentiable nowhere. Show that $f(z)=|z|^2$ is continuous everywhere and differentiable at the origin but nowhere else. 
    
    \newpage
    
    \item [4.7.4]
    
   Let
   
   \[
   f_n(z)=(1+\frac{z}{n})^n
   \]
   Show that 
   
   \[
   f^{\prime}_n(z)=f_{n-1}\left(\frac{(n-1)z}{n}\right)
   \]
   What do you notice as $n\rightarrow \infty$?
    \newpage
    
    \item [4.7.6] Let $f(z)$ be a polynomial in $z\in \mathbb{C}$. Prove that the function $g(z)=\overline{f({\overline{z}})}$
is differentiable everywhere, but that $h(z)=\overline{f(z)}$ is differentiable at 0 if and only if $f^{\prime}(0)=0$. 
    
    \newpage
    \item [4.7.7] In each of the following cases, for $f$ defined on the domain $D$, find the explicit formulas for $u(x,y), v(x,y)$ where $f(z)=u(x,y)+iv(x,y)$, where $z=x+iy$ and all of $x,y,u,v$ are real.
    
    \begin{enumerate}[(i)]
    
        \item $f(z)=1/z, D=\{z\in \mathbb{C}: z\neq 0$.
        
        \item $f(z)=|z|, D= \mathbb{C}$.
        
        \item $f(z)=\overline{z}, D= \mathbb{C}$.
        
        Show that $u,v$ satisfy the Cauchy-Riemann Equations everywhere in $(i)$ and nowhere in $(ii), (iii)$.
        
    \end{enumerate}
    
    \newpage
    \item [4.7.9] 
    For $z=x+iy$, let
    
    \[
    f(z)=\begin{cases} \frac{x^3(1+i)-y^3(1-i)}{x^2+y^2} &\mbox{if } z = 0 \\
0 & \mbox{if } z=0 \end{cases} 
    \]
    Show that $f$ is continuous at the origin, the Cauchy-Riemann Equations are satisfied there, yet $f^{\prime}(0)$ does not exist. Why does this not contradict Theorem 4.12?
\end{enumerate}

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