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{\bf Question}

Find the real and imaginary parts of the function $\sin z$ where
$z=x+iy$.

Describe the images of the lines $y$=constant under the
transformation

$w=\sin z$.  Show that the transformation maps the infinite strip

$$-\frac{\pi}{2}<x<\frac{\pi}{2}, \,\,\,\, y>0$$

conformally onto the upper half plane.

Find a conformal transformation which maps the strip conformally
onto the inside of the unit circle.


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{\bf Answer}

$\sin z=\sin(x+iy)=\sin x\cos iy+\cos x\sin iy$

$\hspace{0.3in}=\sin x\cosh y+i\cos x\sinh iy$

$y={\rm constant\ } \rightarrow w=k_1\sin x+ik_2\cos x$ - ellipse

$x=-\frac{\pi}{2}\Rightarrow w=-\cosh y \hspace{0.2in} y>0$ so
$-\infty<w<-1$ real

$y=0 \Rightarrow w=\sin x \hspace{0.2in} -\frac{\pi}{2}\leq
x\leq\frac{\pi}{2}$ so $-1\leq w\leq1$ real

$x=\frac{\pi}{2}\Rightarrow w=\cosh y \hspace{0.2in} y>0$ so
$1<w<\infty$ real

Thus $w=\sin z$ maps the boundary of $S$ to the real axis.  $z=i
\Rightarrow w=i\sinh1$ which is in $U$.  So $w=\sin z$ maps $S$
conformally onto $U$.

Now $\ds w=\frac{z-i}{z+i}$ maps $U$ to $D$.

So $\ds w=\frac{\sin z-i}{\sin z +i}$ maps $S$ to $D$.

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