Geometric Mapping of the Complex Exponential Function exp(z)
2018-04-09 03:07
This applet visually demonstrates the geometric mapping properties of the complex exponential function w = exp(z). By plotting orthogonal grid lines (horizontal and vertical) in the complex z-plane and showing their images in the w-plane (concentric circles and rays), it helps learners intuitively understand conformal mapping and the geometric significance of complex exponential functions.
In collections
複數
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<show object="true" label="false" ev="7"/>
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<coordStyle style="complex"/>
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<expression label="b1" exp="ℯ^(a1)" type="point" />
<element type="point" label="b1">
<show object="true" label="false" ev="7"/>
<objColor r="255" g="127" b="0" alpha="0"/>
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<expression label="b2" exp="ℯ^(a2)" type="point" />
<element type="point" label="b2">
<show object="false" label="false" ev="7"/>
<objColor r="68" g="68" b="68" alpha="0"/>
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<expression label="b3" exp="ℯ^(a3)" type="point" />
<element type="point" label="b3">
<show object="false" label="false" ev="7"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
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<coords x="-0.7152033087534154" y="0.7132219620671554" z="1"/>
<coordStyle style="complex"/>
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<expression label="b4" exp="ℯ^(a4)" type="point" />
<element type="point" label="b4">
<show object="false" label="false" ev="7"/>
<objColor r="68" g="68" b="68" alpha="0"/>
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<expression label="b5" exp="ℯ^(a5)" type="point" />
<element type="point" label="b5">
<show object="false" label="false" ev="7"/>
<objColor r="68" g="68" b="68" alpha="0"/>
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<expression label="b6" exp="ℯ^(a6)" type="point" />
<element type="point" label="b6">
<show object="true" label="false" ev="7"/>
<objColor r="255" g="255" b="0" alpha="0"/>
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<command name="ExpCurve">
<input a0="a0" a1="a1"/>
<output a0="c1"/>
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<element type="curvecartesian" label="c1">
<show object="true" label="false" ev="7"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="ExpCurve">
<input a0="a1" a1="a2"/>
<output a0="c2"/>
</command>
<element type="curvecartesian" label="c2">
<show object="true" label="false" ev="7"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="ExpCurve">
<input a0="a2" a1="a3"/>
<output a0="c3"/>
</command>
<element type="curvecartesian" label="c3">
<show object="true" label="false" ev="7"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="ExpCurve">
<input a0="a3" a1="a4"/>
<output a0="c4"/>
</command>
<element type="curvecartesian" label="c4">
<show object="true" label="false" ev="7"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="ExpCurve">
<input a0="a4" a1="a5"/>
<output a0="c5"/>
</command>
<element type="curvecartesian" label="c5">
<show object="true" label="false" ev="7"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="ExpCurve">
<input a0="a5" a1="a6"/>
<output a0="c6"/>
</command>
<element type="curvecartesian" label="c6">
<show object="true" label="false" ev="7"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="ExpCurve">
<input a0="a6" a1="a0"/>
<output a0="c0"/>
</command>
<element type="curvecartesian" label="c0">
<show object="true" label="false" ev="7"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="PolyLine">
<input a0="{a0, a1, a2, a3, a4, a5, a6, a0}"/>
<output a0="f"/>
</command>
<element type="polyline" label="f">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<expression label ="c" exp="x^(2) + y^(2) = 1" type="conic"/>
<element type="conic" label="c">
<show object="true" label="false" ev="11"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-1" A3="0" A4="0" A5="0"/>
<eqnStyle style="specific"/>
</element>
<command name="Polygon">
<input a0="{(0, π), (0, (-π)), (x(C11), (-π)), (x(C11), π)}"/>
<output a0="half"/>
</command>
<element type="polygon" label="half">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
</construction>
</geogebra>