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<B> <A ID="tex2html2129"
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<H1><A ID="SECTION001110000000000000000"></A>
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<A ID="sect7.complex"></A>
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<BR>
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Complex numbers
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</H1>
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<P>
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Complex
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<A ID="7761"></A>numbers are written as:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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Z = a + bi
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\end{displaymath}
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-->
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<IMG
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WIDTH="72" HEIGHT="25" BORDER="0"
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SRC="img624.png"
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ALT="\begin{displaymath}
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Z = a + bi
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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where <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> and <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$"> are real numbers and <IMG
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WIDTH="63" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img625.png"
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ALT="$i=\sqrt{-1}$">. (In this book we'll use
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the upper case Roman letters such as <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$Z$"> to denote complex numbers. Real
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numbers appear as lower case Roman or Greek letters, except for
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integer bounds, usually written as <IMG
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WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img86.png"
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ALT="$M$"> or <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">.) Since a complex
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number has two real components, we use a Cartesian plane (in place of a number
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line) to graph it, as shown in Figure <A HREF="#fig07.01">7.1</A>. The quantities <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> and
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<IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$"> are called the
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<A ID="7764"></A><I>real</I>
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and
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<A ID="7766"></A><I>imaginary parts</I> of <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$Z$">, written as:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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a = \mathrm{re}(Z)
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\end{displaymath}
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-->
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<IMG
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WIDTH="65" HEIGHT="28" BORDER="0"
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SRC="img626.png"
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ALT="\begin{displaymath}
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a = \mathrm{re}(Z)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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b = \mathrm{im}(Z)
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\end{displaymath}
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-->
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<IMG
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WIDTH="68" HEIGHT="28" BORDER="0"
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SRC="img627.png"
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ALT="\begin{displaymath}
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b = \mathrm{im}(Z)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<P>
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<DIV ALIGN="CENTER"><A ID="fig07.01"></A><A ID="7772"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.1:</STRONG>
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A number, <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$Z$">, in the complex plane. The axes are for the real
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part <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> and the imaginary part <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$">.</CAPTION>
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<TR><TD><IMG
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WIDTH="254" HEIGHT="205" BORDER="0"
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SRC="img628.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.01.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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If <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$Z$"> is a complex number, its
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<A ID="7775"></A><A ID="7776"></A><I>magnitude</I> (or <I>absolute value</I>),
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written as <IMG
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WIDTH="24" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img629.png"
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ALT="$\vert Z\vert$">, is just the distance in the plane from the origin to the
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point <IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img630.png"
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ALT="$(a,b)$">:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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|Z| = \sqrt {({a^2} + {b^2})}
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\end{displaymath}
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-->
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<IMG
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WIDTH="118" HEIGHT="28" BORDER="0"
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SRC="img631.png"
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ALT="\begin{displaymath}
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\vert Z\vert = \sqrt {({a^2} + {b^2})}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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and its
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<A ID="7781"></A><I>argument</I>,
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written as <IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img632.png"
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ALT="$\angle(Z)$">,
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is the angle from the positive <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> axis to the point <IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img630.png"
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ALT="$(a,b)$">:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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\angle(Z) = \mathrm{arctan}
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{ \left ( {
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b \over a
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} \right ) }
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\end{displaymath}
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-->
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<IMG
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WIDTH="130" HEIGHT="45" BORDER="0"
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SRC="img633.png"
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ALT="\begin{displaymath}
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\angle(Z) = \mathrm{arctan}
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{ \left ( {
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b \over a
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} \right ) }
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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If we know the magnitude and argument of a complex number (call them <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img467.png"
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ALT="$r$"> and
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<IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img634.png"
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ALT="$\theta$">) we can reconstruct the real and imaginary parts:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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a = r \cos (\theta)
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\end{displaymath}
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-->
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<IMG
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WIDTH="79" HEIGHT="28" BORDER="0"
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SRC="img635.png"
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ALT="\begin{displaymath}
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a = r \cos (\theta)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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b = r \sin (\theta)
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\end{displaymath}
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-->
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<IMG
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WIDTH="76" HEIGHT="28" BORDER="0"
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SRC="img636.png"
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ALT="\begin{displaymath}
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b = r \sin (\theta)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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A complex number may be written in terms of its real and imaginary parts
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<IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> and <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$">, as <IMG
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WIDTH="76" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img637.png"
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ALT="$Z=a+bi$"> (this is called
|
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<A ID="7785"></A><I>rectangular form</I>), or alternatively in
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<A ID="7787"></A><I>polar form</I>,
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in terms of <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img467.png"
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ALT="$r$"> and <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img634.png"
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ALT="$\theta$">:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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Z = r \cdot \left[ \cos(\theta) + i \sin(\theta) \right ]
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\end{displaymath}
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-->
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<IMG
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WIDTH="167" HEIGHT="28" BORDER="0"
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SRC="img638.png"
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ALT="\begin{displaymath}
|
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Z = r \cdot \left[ \cos(\theta) + i \sin(\theta) \right ]
|
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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The rectangular and polar formulations are interchangeable; the equations
|
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above show how to compute <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> and <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$"> from <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img467.png"
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ALT="$r$"> and <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img634.png"
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ALT="$\theta$"> and vice versa.
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<P>
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|
The main reason we use complex numbers in electronic music is because they
|
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|
|
magically automate trigonometric calculations. We frequently have to add
|
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|
|
angles together in order to talk about the changing phase of an audio signal as
|
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|
time progresses (or as it is shifted in time, as in this chapter). It turns
|
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out that, if you multiply two complex numbers, the argument of the product is
|
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|
the sum of the arguments of the two factors. To see how this happens, we'll
|
|
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|
|
multiply two numbers <IMG
|
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|
WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
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|
SRC="img639.png"
|
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|
ALT="$Z_1$"> and <IMG
|
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|
WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
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|
|
SRC="img640.png"
|
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|
ALT="$Z_2$">, written in polar form:
|
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|
<BR><P></P>
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|
<DIV ALIGN="CENTER">
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|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
{Z_1} = {r_1} \cdot \left [ \cos({\theta_1}) + i \sin({\theta_1}) \right ]
|
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|
|
\end{displaymath}
|
|
|
|
|
-->
|
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|
|
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|
|
|
<IMG
|
|
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|
|
WIDTH="194" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img641.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
{Z_1} = {r_1} \cdot \left [ \cos({\theta_1}) + i \sin({\theta_1}) \right ]
|
|
|
|
|
\end{displaymath}">
|
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|
|
</DIV>
|
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|
<BR CLEAR="ALL">
|
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|
<P></P>
|
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|
<BR><P></P>
|
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|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
{Z_2} = {r_2} \cdot \left [ \cos({\theta_2}) + i \sin({\theta_2}) \right ]
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="193" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img642.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
{Z_2} = {r_2} \cdot \left [ \cos({\theta_2}) + i \sin({\theta_2}) \right ]
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
giving:
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
{Z_1}{Z_2} = {r_1}{r_2} \cdot
|
|
|
|
|
{ \left [ {
|
|
|
|
|
\cos({\theta_1}) \cos({\theta_2}) -
|
|
|
|
|
\sin({\theta_1}) \sin({\theta_2})
|
|
|
|
|
} \right . } \, +
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="332" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img643.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
{Z_1}{Z_2} = {r_1}{r_2} \cdot
|
|
|
|
|
{ \left [ {
|
|
|
|
|
\cos({\theta_1...
|
|
|
|
|
...a_2}) -
|
|
|
|
|
\sin({\theta_1}) \sin({\theta_2})
|
|
|
|
|
} \right . } \, +
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
{ \left . {
|
|
|
|
|
+ i \left (
|
|
|
|
|
\sin({\theta_1}) \cos({\theta_2}) +
|
|
|
|
|
\cos({\theta_1}) \sin({\theta_2})
|
|
|
|
|
\right )
|
|
|
|
|
} \right ] }
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="249" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img644.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
{ \left . {
|
|
|
|
|
+ i \left (
|
|
|
|
|
\sin({\theta_1}) \cos({\theta_2}) +
|
|
|
|
|
\cos({\theta_1}) \sin({\theta_2})
|
|
|
|
|
\right )
|
|
|
|
|
} \right ] }
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
Here the minus sign in front of the <!-- MATH
|
|
|
|
|
$\sin({\theta_1}) \sin({\theta_2})$
|
|
|
|
|
-->
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="99" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img645.png"
|
|
|
|
|
ALT="$\sin({\theta_1}) \sin({\theta_2})$"> term
|
|
|
|
|
comes from multiplying <IMG
|
|
|
|
|
WIDTH="9" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img646.png"
|
|
|
|
|
ALT="$i$"> by itself, which gives <IMG
|
|
|
|
|
WIDTH="23" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img401.png"
|
|
|
|
|
ALT="$-1$">. We can spot the
|
|
|
|
|
cosine and sine summation formulas in the above expression, and so it simplifies
|
|
|
|
|
to:
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
{Z_1}{Z_2} = {r_1}{r_2}
|
|
|
|
|
\cdot \left[ \cos({\theta_1}+{\theta_2})
|
|
|
|
|
+ i \sin({\theta_1}+{\theta_2}) \right ]
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="294" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img647.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
{Z_1}{Z_2} = {r_1}{r_2}
|
|
|
|
|
\cdot \left[ \cos({\theta_1}+{\theta_2})
|
|
|
|
|
+ i \sin({\theta_1}+{\theta_2}) \right ]
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
By inspection, it follows that the product <IMG
|
|
|
|
|
WIDTH="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img648.png"
|
|
|
|
|
ALT="${Z_1}{Z_2}$">
|
|
|
|
|
has magnitude
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="32" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img649.png"
|
|
|
|
|
ALT="${r_1}{r_2}$"> and argument <!-- MATH
|
|
|
|
|
${\theta_1}+{\theta_2}$
|
|
|
|
|
-->
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img650.png"
|
|
|
|
|
ALT="${\theta_1}+{\theta_2}$">.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
We can use this property of complex numbers to add and subtract angles (by
|
|
|
|
|
multiplying and dividing complex numbers with the appropriate arguments) and
|
|
|
|
|
then to take the cosine and sine of the result by extracting the real and
|
|
|
|
|
imaginary parts.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
<BR><HR>
|
|
|
|
|
<!--Table of Child-Links-->
|
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|
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<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
2022-04-12 21:54:18 -03:00
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<UL>
|
2022-04-12 23:32:40 -03:00
|
|
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<LI><A ID="tex2html2132"
|
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HREF="node106.html">Complex sinusoids</A>
|
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HREF="node106.html">
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HREF="node106.html">Complex sinusoids</A>
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<B> Up:</B> <A ID="tex2html2125"
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HREF="node104.html">Time shifts and delays</A>
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HREF="node104.html">Time shifts and delays</A>
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<B> <A ID="tex2html2127"
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HREF="node4.html">Contents</A></B>
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<B> <A ID="tex2html2129"
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