39. Show that the mapping $z=w+e^{w}$ is conformal except at a finite set of points.

A parallel plate capacitor has plates that extend from $x=-1$ to $x=-\infty $. Find an appropriate scaling that allows you to place the plates at $y=\pm \pi .$ Show that the given transformation maps the plates to the lines $v=\pm \pi .$ Solve for the potential between the plates in the $w-$plane, map to the $z-$plane$,$ and hence find the equipotential surfaces at the ends of the capacitor. Sketch the field lines. This is the so-called fringing field.

Choose $y=s\pi /2d$ where $s$ is a coordinate measured perpendicular to the plates, and $d$ is the plate separation. The function $w+e^{w}$ is analytic everywhere, and the derivative is
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is non-zero except at the points
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or, equivalently,
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The mapping takes the form:
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Then for $v=0$ MATH $x $ ranges from $-\infty $ to $+\infty ,$ i.e. we get the whole real axis in the $z-$plane. The line $\omega =i\pi $ maps to $x=u-e^{u},$ $y=\pi .$ $x$ ranges from $-\infty $ at $u=\infty $ to $-1$ at $u=0.$ This is the top plate of the capacitor. Similarly $w=-i\pi $ maps to the lower plate.

The mapping MATH has a branch point at each of the points MATH Each 2$\pi -$wide strip of the $w-$plane maps to the whole $z-$plane$.$ For each branch there are two points in the $z-$plane at which the mapping is not conformal.

In the $w-$plane we can write the potential as $\phi =vV/2\pi ,$ giving a complex potential $\Phi =wV/2\pi ,$ with the complex part being the physical potential. Equipotentials correspond to $v=$const $=v_{0}.$ The corresponding curves in the $z-$plane are:
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Thus
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and
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The equipotenials are shown in the figure.
Ch2pr39__48.png

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