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\begin{center}
{\huge  \sc  Math 122A Homework 4 } \\
\smallskip
{Due date: 2/4/21}\\
\smallskip
{Name: }\\
{\large  }
\end{center}

\begin{enumerate}
\item[4.7.11] Consider 
\[
f(z)=\begin{cases} \frac{xy^2(x+iy)}{x^2+x^4} &\mbox{if } z=x+iy \neq 0 \\
0 & \mbox{if } z=0 \end{cases} 
\]
Verify that $\lim_{z\rightarrow z_0}(f(z)-f(z_0))/(z-z_0)=0$ as $z\rightarrow 0$ along any straight line, $z=(a+ib)t, t\in \mathbb{R}$. This does not prove that $f^{\prime}(z)=0$, however. By considering $z\rightarrow 0$ along the path$z(t)=t^2+it$, show that $f$ is not differentiable at 0.( This shows that when computing $f^{\prime}$ it is not enough to consider a limit taken along certain specific paths or types of path. An entire neighbourhood of the point concerned must be considered.)
\newpage

    \item [4.7.12] For each of the following, compute $f^{\prime}(\gamma(t)), \gamma^{\prime}(t), (f\gamma)^{\prime}(t)$ and verify directly that 
    \[
    (f\gamma)^{\prime}(t)=f^{\prime}(\gamma(t)) \gamma^{\prime}(t)
    \]
    
   \begin{enumerate}[(i)]
       
        
        \item $f(z)=z^2, \gamma(t)=t^3+it^4 \ (z\in \mathbb{C}, t\in[0,1] $
        
         \item $f(z)=\frac{1}{z}, \gamma(t)=\cos(t)+i\sin(t) \ (z\neq 0, t\in[0,2\pi]) $
         
         \item $f(z)=1+z+z^2+\ldots , \gamma(t)=t+it^2 \ (|z|<1, t\in[0,\frac{1}{2}])$
          
          
        
    \end{enumerate}

    
    \newpage
    
    \item [4.7.15]  Assume that the two power series $s(z)=\sum a_nz^n$ and $c(z)=\sum b_nz^n$ are convergent for all $z\in\mathbb{C}$, and that they satisfy the relations $s^{\prime}(z)=c(z)$, $c^{\prime}(z)=-s(z)$. Deduce the identities
    \[
    a_n=-a_{n-2}/(n(n-1)) \qquad b_n=-a_{n-2}/(n(n-1))
    \]
    If further $s(0)=0$, $c(0)=1$, determine $s(z)$ and $c(z)$ completely. By differentiation, or otherwise, prove that
    \[
    (c(z))^2+(s(z))^2=1.
    \]
    \newpage
    
    \item[5.10.1] Express the following in the form $a+bi$ for real $a,b:$
    \begin{enumerate}[(i)]
        \item $e^i$
        
        \item $e^{2+\pi i}$
        
        \item $1/e^{2+\pi i}$.
        
    \end{enumerate}
    
\newpage

\item[5.10.5] Use the identity $e^{i\theta}e^{i\phi}=e^{i(\theta +\phi}$ to derive the usual formula for $\cos(\theta +\phi)$ and $\sin(\theta + \phi)$. By a similar method, show that $1/(\cos(\theta)+i\sin(\theta))=\cos(\theta)-i\sin(\theta)$.

\newpage

\item [5.10.15] By considering the real and imaginary parts of 
\[
1+z+z^2+ \ldots + z^n=\frac{1-z^{n+1}}{1-z}
\]
Find the explicit formulas for the sums:
\begin{enumerate}[(i)]

\item $\sum_{m=0}^{n}\cos(mx)$

\item $\sum_{m=0}^{n}\sin(mx)$
\smallskip

By similar methods, find:

\item  $\sum_{m=1}^{n}\cos((2m-1)x)$

\item  $\sum_{m=1}^{n}\sin((2m-1)x)$

\item  $\sum_{m=1}^{n}(-1)^n\sin(mx)$

\item  $\sum_{m=0}^{n}\cos(\theta +m\phi)$

\item  $\sum_{m=0}^{n}\sin(\theta +m\phi)$


\end{enumerate}

\newpage
    
    \item [6.14.1] Draw the contours $\gamma=[0,i]$, $\sigma=[0,1]+[1,i]$ Evaluate
    \[
    \int_{\gamma}re(z)dz \qquad \int_{\sigma}re(z)dz
    \]
    
    \newpage
    
    \item [6.14.2]
    
    Draw the contours $\gamma=[-i,i]$ and $\sigma$ where $\sigma(t)=e^{it} (t\in[-\pi/2,\pi/2]).$ Evaluate
    \[
    \int_{\gamma}|z|dz \qquad \int_{\sigma}|z|dz
    \]
    \newpage
    
    \item [6.14.4]
    
   Let $n$ be a positive integer. If $\gamma(t)=z_0+re^{it} \ (t\in[0,2n\pi])$ describe the contours $\gamma$ and $-\gamma$ geometrically. Compute
   \[
   \int_{\gamma}\frac{1}{z-z_0}dz \ \mbox{ and } \ \int_{-\gamma}\frac{1}{z-z_0}dz
   \]
    \newpage
    
    \item [6.14.5] For $\gamma(t)=e^{it} (t\in[0,\pi])$ evaluate $\int_{\gamma}f$ for each of the following functions $f$:
    
    \begin{enumerate}[(i)]
        \item $1/z^2$
        \item $1/z$
        \item $\cos(z)$
        \item $\sinh(z)$
        \item $\tan(z)$
        \item $e^{z^3}$
        
    \end{enumerate}
    
    \newpage
    \item [6.14.12] Let $c_r(t)=z_0+re^{it} (t\in[0,2\pi])$ be the circle centre $z_0$, radius $r>0$. If $f$ is continuous in the domain $D$ and $z_0\in D$, use the Estimation Lemma to prove
    
    
    \begin{enumerate}[(i)]
    
        \item $\lim_{r\rightarrow 0} \int_{c_{r}}f(z)dz=0$
        
        \item $\lim_{r\rightarrow 0} \int_{c_{r}}\dfrac{f(z)}{z-z_0}dz=2\pi i f(z_0)$
        
        
        
    \end{enumerate}
    
    \newpage
    \item [6.14.13] 
    For each of the following functions $f: D \rightarrow \mathbb{C}$, either state an antiderivative $F: D\rightarrow \mathbb{C}$ or explain why an antiderivative does not exist. 
    
    \begin{enumerate}[(i)]
    
        \item $f(z)=z^2, \ D=\mathbb{C}$
        \item $f(z)=1/z^2, \ D=\mathbb{C}\setminus\{0\}$
        \item $f(z)=1/z, \ D=\mathbb{C}\setminus\{0\}$
        \item $f(z)=z\sin(z),\  D=\mathbb{C}$
        \item $f(z)=|z|^2, \ D=\mathbb{C}$
        \item $f(z)=\overline{z},\  D=\mathbb{C}$
        
        
        
    \end{enumerate}
\end{enumerate}

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