<?xml version="1.0" encoding="UTF-8"?>
<?latexml class="article"?>
<?latexml package="cases"?>
<?latexml RelaxNGSchema="LaTeXML"?>
<document xmlns="http://dlmf.nist.gov/LaTeXML">
<resource src="LaTeXML.css" type="text/css"/>
<resource src="ltx-article.css" type="text/css"/>
<para xml:id="p1">
<equationgroup class="ltx_eqn_numcases" xml:id="S0.EGx1">
<equation xml:id="S0.E1">
<tags>
<tag>(1)</tag>
<tag role="refnum">1</tag>
</tags>
<MathFork>
<Math tex="\displaystyle|x|=x,for x\geq 0" text="formulae@(absolute-value@(x) = x, [for ] * x >= 0)" xml:id="S0.E1.m4">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="formulae"/>
<XMRef idref="S0.E1.m4.2"/>
<XMRef idref="S0.E1.m4.3"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S0.E1.m4.2">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E1.m4.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E1.m4.1">x</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
<XMTok role="PUNCT">,</XMTok>
<XMApp xml:id="S0.E1.m4.3">
<XMTok meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText>for </XMText>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMWrap>
</XMDual>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle|x|=" text="absolute-value@(x) = absent" xml:id="S0.E1.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E1.m1.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E1.m1.1">x</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMTok meaning="absent"/>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle x," text="x" xml:id="S0.E1.m2">
<XMath>
<XMDual>
<XMRef idref="S0.E1.m2.1"/>
<XMWrap>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E1.m2.1">x</XMTok>
<XMTok role="PUNCT">,</XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math></td>
<td align="left">for <Math mode="inline" tex="x\geq 0" text="x >= 0" xml:id="S0.E1.m3">
<XMath>
<XMApp>
<XMTok meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
<equation xml:id="S0.E2">
<tags>
<tag>(2)</tag>
<tag role="refnum">2</tag>
</tags>
<MathFork>
<Math tex="\displaystyle|x|=-x,for x<0" text="formulae@(absolute-value@(x) = - x, [for ] * x < 0)" xml:id="S0.E2.m4">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="formulae"/>
<XMRef idref="S0.E2.m4.2"/>
<XMRef idref="S0.E2.m4.3"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S0.E2.m4.2">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E2.m4.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E2.m4.1">x</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMApp>
<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
<XMTok role="PUNCT">,</XMTok>
<XMApp xml:id="S0.E2.m4.3">
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText>for </XMText>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMWrap>
</XMDual>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle|x|=" text="absolute-value@(x) = absent" xml:id="S0.E2.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E2.m1.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E2.m1.1">x</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMTok meaning="absent"/>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle-x," text="- x" xml:id="S0.E2.m2">
<XMath>
<XMDual>
<XMRef idref="S0.E2.m2.1"/>
<XMWrap>
<XMApp xml:id="S0.E2.m2.1">
<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
<XMTok role="PUNCT">,</XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math></td>
<td align="left">for <Math mode="inline" tex="x<0" text="x < 0" xml:id="S0.E2.m3">
<XMath>
<XMApp>
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
</equationgroup>
</para>
<para xml:id="p2">
<p>Another example is calculating the square root of <Math mode="inline" tex="c+id" text="c + i * d" xml:id="p2.m1">
<XMath>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math>. First compute</p>
<equationgroup class="ltx_eqn_numcases" xml:id="S0.EGx2">
<equation labels="LABEL:w LABEL:wzero" xml:id="S0.E3.1">
<tags>
<tag>(3a)</tag>
<tag role="refnum">3a</tag>
</tags>
<MathFork>
<Math tex="\displaystyle w\equiv 0c=d=0" text="w equivalent-to 0 * c = d = 0" xml:id="S0.E3.1.m4">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok meaning="equivalent-to" name="equiv" role="RELOP">≡</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle w\equiv" text="w equivalent-to absent" xml:id="S0.E3.1.m1">
<XMath>
<XMApp>
<XMTok meaning="equivalent-to" name="equiv" role="RELOP">≡</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok meaning="absent"/>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle 0" text="0" xml:id="S0.E3.1.m2">
<XMath>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="c=d=0" text="c = d = 0" xml:id="S0.E3.1.m3">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
<equation labels="LABEL:w" xml:id="S0.E3.2">
<tags>
<tag>(3b)</tag>
<tag role="refnum">3b</tag>
</tags>
<MathFork>
<Math tex="\displaystyle w\equiv\sqrt{|c|}\,\sqrt{\frac{1+\sqrt{1+(d/c)^{2}}}{2}}|c|\geq|d|" text="w equivalent-to square-root@(absolute-value@(c)) * square-root@((1 + square-root@(1 + (d / c) ^ 2)) / 2) * absolute-value@(c) >= absolute-value@(d)" xml:id="S0.E3.2.m4">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok meaning="equivalent-to" name="equiv" role="RELOP">≡</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMApp rpadding="1.7pt">
<XMTok meaning="square-root"/>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.2.m4.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m4.1">c</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok meaning="1" role="NUMBER">1</XMTok>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok meaning="1" role="NUMBER">1</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMDual>
<XMRef idref="S0.E3.2.m4.2"/>
<XMWrap>
<XMTok role="OPEN" stretchy="false">(</XMTok>
<XMApp xml:id="S0.E3.2.m4.2">
<XMTok meaning="divide" role="MULOP">/</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
<XMTok role="CLOSE" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.2.m4.3"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m4.3">c</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
<XMTok meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.2.m4.4"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m4.4">d</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle w\equiv" text="w equivalent-to absent" xml:id="S0.E3.2.m1">
<XMath>
<XMApp>
<XMTok meaning="equivalent-to" name="equiv" role="RELOP">≡</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok meaning="absent"/>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle\sqrt{|c|}\,\sqrt{\frac{1+\sqrt{1+(d/c)^{2}}}{2}}" text="square-root@(absolute-value@(c)) * square-root@((1 + square-root@(1 + (d / c) ^ 2)) / 2)" xml:id="S0.E3.2.m2">
<XMath>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMApp rpadding="1.7pt">
<XMTok meaning="square-root"/>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.2.m2.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m2.1">c</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok meaning="1" role="NUMBER">1</XMTok>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok meaning="1" role="NUMBER">1</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMDual>
<XMRef idref="S0.E3.2.m2.2"/>
<XMWrap>
<XMTok role="OPEN" stretchy="false">(</XMTok>
<XMApp xml:id="S0.E3.2.m2.2">
<XMTok meaning="divide" role="MULOP">/</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
<XMTok role="CLOSE" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="|c|\geq|d|" text="absolute-value@(c) >= absolute-value@(d)" xml:id="S0.E3.2.m3">
<XMath>
<XMApp>
<XMTok meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.2.m3.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m3.1">c</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.2.m3.2"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m3.2">d</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
<equation labels="LABEL:w" xml:id="S0.E3.3">
<tags>
<tag>(3c)</tag>
<tag role="refnum">3c</tag>
</tags>
<MathFork>
<Math tex="\displaystyle w\equiv\sqrt{|d|}\,\sqrt{\frac{|c/d|+\sqrt{1+(c/d)^{2}}}{2}}|c|<% |d|" text="w equivalent-to square-root@(absolute-value@(d)) * square-root@((absolute-value@(c / d) + square-root@(1 + (c / d) ^ 2)) / 2) * absolute-value@(c) < absolute-value@(d)" xml:id="S0.E3.3.m4">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok meaning="equivalent-to" name="equiv" role="RELOP">≡</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMApp rpadding="1.7pt">
<XMTok meaning="square-root"/>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.3.m4.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m4.1">d</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.3.m4.3"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMApp xml:id="S0.E3.3.m4.3">
<XMTok meaning="divide" role="MULOP">/</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok meaning="1" role="NUMBER">1</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMDual>
<XMRef idref="S0.E3.3.m4.2"/>
<XMWrap>
<XMTok role="OPEN" stretchy="false">(</XMTok>
<XMApp xml:id="S0.E3.3.m4.2">
<XMTok meaning="divide" role="MULOP">/</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
<XMTok role="CLOSE" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.3.m4.4"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m4.4">c</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.3.m4.5"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m4.5">d</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle w\equiv" text="w equivalent-to absent" xml:id="S0.E3.3.m1">
<XMath>
<XMApp>
<XMTok meaning="equivalent-to" name="equiv" role="RELOP">≡</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok meaning="absent"/>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle\sqrt{|d|}\,\sqrt{\frac{|c/d|+\sqrt{1+(c/d)^{2}}}{2}}" text="square-root@(absolute-value@(d)) * square-root@((absolute-value@(c / d) + square-root@(1 + (c / d) ^ 2)) / 2)" xml:id="S0.E3.3.m2">
<XMath>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMApp rpadding="1.7pt">
<XMTok meaning="square-root"/>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.3.m2.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m2.1">d</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.3.m2.3"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMApp xml:id="S0.E3.3.m2.3">
<XMTok meaning="divide" role="MULOP">/</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok meaning="1" role="NUMBER">1</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMDual>
<XMRef idref="S0.E3.3.m2.2"/>
<XMWrap>
<XMTok role="OPEN" stretchy="false">(</XMTok>
<XMApp xml:id="S0.E3.3.m2.2">
<XMTok meaning="divide" role="MULOP">/</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
<XMTok role="CLOSE" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="|c|<|d|" text="absolute-value@(c) < absolute-value@(d)" xml:id="S0.E3.3.m3">
<XMath>
<XMApp>
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.3.m3.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m3.1">c</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E3.3.m3.2"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m3.2">d</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
</equationgroup>
<p>Then, using <Math mode="inline" tex="w" text="w" xml:id="p2.m2">
<XMath>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMath>
</Math> from eq. (<ref labelref="LABEL:w"/>), the square root is</p>
<equationgroup class="ltx_eqn_numcases" xml:id="S0.EGx3">
<equation xml:id="S0.E4.1">
<tags>
<tag>(4a)</tag>
<tag role="refnum">4a</tag>
</tags>
<MathFork>
<Math tex="\displaystyle\sqrt{c+id}=0w=0 (case )" text="square-root@(c + i * d) = 0 * w = 0 * [ (case ] * [)]" xml:id="S0.E4.1.m4">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
<XMText> (case </XMText>
<XMText>)</XMText>
</XMApp>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle\sqrt{c+id}=" text="square-root@(c + i * d) = absent" xml:id="S0.E4.1.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="absent"/>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle 0" text="0" xml:id="S0.E4.1.m2">
<XMath>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="w=0" text="w = 0" xml:id="S0.E4.1.m3">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math> (case <ref labelref="LABEL:wzero"/>)</td>
</MathBranch>
</MathFork>
</equation>
<equation xml:id="S0.E4.2">
<tags>
<tag>(4b)</tag>
<tag role="refnum">4b</tag>
</tags>
<MathFork>
<Math tex="\displaystyle\sqrt{c+id}=w+i\frac{d}{2w}w\neq 0, c\geq 0" text="square-root@(c + i * d) = w + i * (d / (2 * w)) * w not-equals 0 * [, ] * c >= 0" xml:id="S0.E4.2.m5">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="2" role="NUMBER">2</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
</XMApp>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
</XMApp>
<XMTok meaning="not-equals" name="neq" role="RELOP">≠</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
<XMText>, </XMText>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
<XMTok meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle\sqrt{c+id}=" text="square-root@(c + i * d) = absent" xml:id="S0.E4.2.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="absent"/>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle w+i\frac{d}{2w}" text="w + i * (d / (2 * w))" xml:id="S0.E4.2.m2">
<XMath>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="2" role="NUMBER">2</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="w\neq 0" text="w not-equals 0" xml:id="S0.E4.2.m3">
<XMath>
<XMApp>
<XMTok meaning="not-equals" name="neq" role="RELOP">≠</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math>, <Math mode="inline" tex="c\geq 0" text="c >= 0" xml:id="S0.E4.2.m4">
<XMath>
<XMApp>
<XMTok meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
<equation xml:id="S0.E4.3">
<tags>
<tag>(4c)</tag>
<tag role="refnum">4c</tag>
</tags>
<MathFork>
<Math tex="\displaystyle\sqrt{c+id}=\frac{|d|}{2w}+iww\neq 0, c<0, d\geq 0" text="square-root@(c + i * d) = absolute-value@(d) / (2 * w) + i * w * w not-equals 0 * [, ] * c < 0 * [, ] * d >= 0" xml:id="S0.E4.3.m6">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E4.3.m6.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E4.3.m6.1">d</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="2" role="NUMBER">2</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
</XMApp>
<XMTok meaning="not-equals" name="neq" role="RELOP">≠</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
<XMText>, </XMText>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
<XMText>, </XMText>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
<XMTok meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle\sqrt{c+id}=" text="square-root@(c + i * d) = absent" xml:id="S0.E4.3.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="absent"/>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle\frac{|d|}{2w}+iw" text="absolute-value@(d) / (2 * w) + i * w" xml:id="S0.E4.3.m2">
<XMath>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E4.3.m2.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E4.3.m2.1">d</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="2" role="NUMBER">2</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="w\neq 0" text="w not-equals 0" xml:id="S0.E4.3.m3">
<XMath>
<XMApp>
<XMTok meaning="not-equals" name="neq" role="RELOP">≠</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math>, <Math mode="inline" tex="c<0" text="c < 0" xml:id="S0.E4.3.m4">
<XMath>
<XMApp>
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math>, <Math mode="inline" tex="d\geq 0" text="d >= 0" xml:id="S0.E4.3.m5">
<XMath>
<XMApp>
<XMTok meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
<equation xml:id="S0.E4.4">
<tags>
<tag>(4d)</tag>
<tag role="refnum">4d</tag>
</tags>
<MathFork>
<Math tex="\displaystyle\sqrt{c+id}=\frac{|d|}{2w}-iww\neq 0, c<0, d<0" text="square-root@(c + i * d) = absolute-value@(d) / (2 * w) - i * w * w not-equals 0 * [, ] * c < 0 * [, ] * d < 0" xml:id="S0.E4.4.m6">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E4.4.m6.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E4.4.m6.1">d</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="2" role="NUMBER">2</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
</XMApp>
<XMTok meaning="not-equals" name="neq" role="RELOP">≠</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
<XMText>, </XMText>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
<XMText>, </XMText>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle\sqrt{c+id}=" text="square-root@(c + i * d) = absent" xml:id="S0.E4.4.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="absent"/>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle\frac{|d|}{2w}-iw" text="absolute-value@(d) / (2 * w) - i * w" xml:id="S0.E4.4.m2">
<XMath>
<XMApp>
<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S0.E4.4.m2.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S0.E4.4.m2.1">d</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="2" role="NUMBER">2</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="w\neq 0" text="w not-equals 0" xml:id="S0.E4.4.m3">
<XMath>
<XMApp>
<XMTok meaning="not-equals" name="neq" role="RELOP">≠</XMTok>
<XMTok font="italic" role="UNKNOWN">w</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math>, <Math mode="inline" tex="c<0" text="c < 0" xml:id="S0.E4.4.m4">
<XMath>
<XMApp>
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math>, <Math mode="inline" tex="d<0" text="d < 0" xml:id="S0.E4.4.m5">
<XMath>
<XMApp>
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
</equationgroup>
</para>
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