<?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 &gt;= 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="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E1.m4.1">x</XMTok>
                        <XMTok role="VERTBAR" 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="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E1.m1.1">x</XMTok>
                        <XMTok role="VERTBAR" 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 &gt;= 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&lt;0" text="formulae@(absolute-value@(x) = - x, [for ] * x &lt; 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="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E2.m4.1">x</XMTok>
                        <XMTok role="VERTBAR" 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">&lt;</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="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E2.m1.1">x</XMTok>
                        <XMTok role="VERTBAR" 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&lt;0" text="x &lt; 0" xml:id="S0.E2.m3">
                <XMath>
                  <XMApp>
                    <XMTok meaning="less-than" role="RELOP">&lt;</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) &gt;= 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="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m4.1">c</XMTok>
                        <XMTok role="VERTBAR" 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="VERTBAR" stretchy="false">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m4.3">c</XMTok>
                      <XMTok role="VERTBAR" 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="VERTBAR" stretchy="false">|</XMTok>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m4.4">d</XMTok>
                    <XMTok role="VERTBAR" 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="VERTBAR" stretchy="false">|</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m2.1">c</XMTok>
                          <XMTok role="VERTBAR" 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) &gt;= 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="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m3.1">c</XMTok>
                        <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                      </XMWrap>
                    </XMDual>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="absolute-value"/>
                        <XMRef idref="S0.E3.2.m3.2"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.2.m3.2">d</XMTok>
                        <XMTok role="VERTBAR" 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|&lt;%&#10;|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) &lt; 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="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m4.1">d</XMTok>
                        <XMTok role="VERTBAR" 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="VERTBAR" 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="VERTBAR" 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="VERTBAR" stretchy="false">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m4.4">c</XMTok>
                      <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMApp>
                <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="absolute-value"/>
                    <XMRef idref="S0.E3.3.m4.5"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m4.5">d</XMTok>
                    <XMTok role="VERTBAR" 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="VERTBAR" stretchy="false">|</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m2.1">d</XMTok>
                          <XMTok role="VERTBAR" 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="VERTBAR" 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="VERTBAR" 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|&lt;|d|" text="absolute-value@(c) &lt; absolute-value@(d)" xml:id="S0.E3.3.m3">
                <XMath>
                  <XMApp>
                    <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="absolute-value"/>
                        <XMRef idref="S0.E3.3.m3.1"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m3.1">c</XMTok>
                        <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                      </XMWrap>
                    </XMDual>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="absolute-value"/>
                        <XMRef idref="S0.E3.3.m3.2"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E3.3.m3.2">d</XMTok>
                        <XMTok role="VERTBAR" 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 &gt;= 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 &gt;= 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&lt;0, d\geq 0" text="square-root@(c + i * d) = absolute-value@(d) / (2 * w) + i * w * w not-equals 0 * [, ] * c &lt; 0 * [, ] * d &gt;= 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="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E4.3.m6.1">d</XMTok>
                        <XMTok role="VERTBAR" 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">&lt;</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="VERTBAR" stretchy="false">|</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S0.E4.3.m2.1">d</XMTok>
                          <XMTok role="VERTBAR" 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&lt;0" text="c &lt; 0" xml:id="S0.E4.3.m4">
                <XMath>
                  <XMApp>
                    <XMTok meaning="less-than" role="RELOP">&lt;</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 &gt;= 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&lt;0, d&lt;0" text="square-root@(c + i * d) = absolute-value@(d) / (2 * w) - i * w * w not-equals 0 * [, ] * c &lt; 0 * [, ] * d &lt; 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="VERTBAR" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S0.E4.4.m6.1">d</XMTok>
                        <XMTok role="VERTBAR" 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">&lt;</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">&lt;</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="VERTBAR" stretchy="false">|</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S0.E4.4.m2.1">d</XMTok>
                          <XMTok role="VERTBAR" 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&lt;0" text="c &lt; 0" xml:id="S0.E4.4.m4">
                <XMath>
                  <XMApp>
                    <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
                    <XMTok font="italic" role="UNKNOWN">c</XMTok>
                    <XMTok meaning="0" role="NUMBER">0</XMTok>
                  </XMApp>
                </XMath>
              </Math>, <Math mode="inline" tex="d&lt;0" text="d &lt; 0" xml:id="S0.E4.4.m5">
                <XMath>
                  <XMApp>
                    <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
                    <XMTok font="italic" role="UNKNOWN">d</XMTok>
                    <XMTok meaning="0" role="NUMBER">0</XMTok>
                  </XMApp>
                </XMath>
              </Math></td>
          </MathBranch>
        </MathFork>
      </equation>
    </equationgroup>
  </para>
</document>
