Amanda

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Assignment 2: Functions

find the real and imaginary parts of the following functions in terms of polar coordinates r and $\theta$

3. a)
$w=z^3$

(1)
\begin{align} z^3=\left ( re^{i\theta} \right )^3 \end{align}
(2)
\begin{align} =r^3e^{i3\theta}=r^3\left ( cos3\theta+isin3\theta \right ) \end{align}

real and imaginary parts are:

(3)
\begin{align} u=r^3cos3\theta \end{align}
(4)
\begin{align} v=r^3sin3\theta \end{align}

3. b)
$w=z+\frac{1}{z}$

(5)
\begin{align} z+\frac{1}{z}=re^{i\theta}+\frac{1}{re^{i\theta}} \end{align}
(6)
\begin{align} =rcos\theta +irsin\theta+\frac{1}{rcos\theta+irsin\theta} \end{align}

rationalizing the denominator of the fraction:

(7)
\begin{align} \left ( \frac{1}{rcos\theta+irsin\theta} \right )\left ( \frac{rcos\theta-irsin\theta}{rcos\theta-irsin\theta} \right )=\frac{rcos\theta-irsin\theta}{r^2cos^2\theta-ir^2cos{\theta}sin{\theta}+ir^2cos{\theta}sin{\theta}-i^2r^2sin^2\theta} \end{align}
(8)
\begin{align} =\frac{rcos\theta-irsin\theta}{r^2cos^2\theta+r^2sin^2\theta}=\frac{r\left ( cos\theta-isin\theta \right )}{r^2\left ( cos^2\theta+sin^2\theta \right )}=\frac{cos\theta-isin\theta}{r} \end{align}

back to original problem:

(9)
\begin{align} =r\left ( cos\theta+isin\theta \right )+\frac{cos\theta-isin\theta}{r}=\frac{r^2cos\theta+ir^2sin\theta+cos\theta-isin\theta}{r} \end{align}

real and imaginary parts are:

(10)
\begin{align} u=\frac{r^2cos\theta+cos\theta}{r} \end{align}
(11)
\begin{align} v=\frac{r^2sin\theta-sin\theta}{r} \end{align}

Assignment 5: Series

find the Laurent Series

2. b)
$\frac{1}{z(z-3)}$ on the annulus $0<\left | z \right |<3$

(12)
\begin{align} \left ( \frac{1}{z} \right )\left ( \frac{1}{z-3} \right )= \left ( \frac{-1}{3z} \right )\left ( \frac{1}{1-\frac{z}{3}} \right ) \end{align}
(13)
\begin{align} \left ( \frac{-1}{3z} \right )\left ( \frac{1}{1-\frac{z}{3}} \right )= \left ( \frac{-1}{3z} \right )\left ( 1+\frac{z}{3}+\frac{z^2}{9}+\frac{z^3}{27}+... \right ) \end{align}
(14)
\begin{align} =\left ( \frac{-1}{3z}-\frac{1}{9}-\frac{z}{27}+\frac{z^2}{81}+... \right ) \end{align}

Assignment 7: Residues

compute the residue at the poles of the functions

1. a)
$\int _c\frac{sinz}{z^2-1}dz$ c is the circle $\left | z \right |=2$

(15)
\begin{equation} z^2-1=0 \end{equation}
(16)
\begin{align} z=\pm 1 \end{align}

Res$\left ( -1 \right )=\frac{sinz}{2z}$
evaluate $\frac{sinz}{2z}$ at $z=-1$

(17)
\begin{align} \frac{sin\left ( -1 \right )}{2\left ( -1 \right )}=\frac{-.84}{-2}=.42074 \end{align}

Res$\left ( 1 \right )=\frac{sinz}{2z}$
evaluate $\frac{sinz}{2z}$ at $z=1$

(18)
\begin{align} \frac{sin\left ( 1 \right )}{2\left ( 1 \right )}=\frac{.84}{2}=.42074 \end{align}
(19)
\begin{align} 2\pi i\left ( .42074+.42074 \right )=1.68\pi i \end{align}
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