Example: Symbolic Elliptic Integral Functions

Symbolics > Working with Symbolics > Calculus > Example: Symbolic Elliptic Integral Functions

The following elliptic integral functions appear in many symbolic calculations.

The elliptic integral functions are not part of the PTC Mathcad Prime set of built-in functions.

EllipticK: The Complete Elliptic Integral of the First Kind

1. Show the definition of the complete elliptic integral of the first kind, EllipticK(m).

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  <math resultRef="34">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="*" xml:space="preserve">EllipticK</id>
        <boundVars>
          <id labels="*" xml:space="preserve">m</id>
        </boundVars>
      </function>
      <apply>
        <integral />
        <lambda>
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            <id xml:space="preserve">τ</id>
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            <div />
            <real>1</real>
            <apply>
              <mult />
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <minus />
                  <real>1</real>
                  <apply>
                    <pow />
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                    <real>2</real>
                  </apply>
                </apply>
              </apply>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <minus />
                  <real>1</real>
                  <apply>
                    <mult />
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m</id>
                    <apply>
                      <pow />
                      <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                      <real>2</real>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <real>1</real>
        </upperBound>
      </apply>
    </define>
  </math>
</region>

2. Evaluate EllipticK numerically.

3. Plot the numerical values of EllipticK in the range of 0<m<1.

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              <apply xmlns="http://schemas.mathsoft.com/math50">
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                <apply>
                  <div />
                  <id xml:space="preserve">l</id>
                  <real>10</real>
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                  <real>400</real>
                </apply>
              </math>
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                    <real>400</real>
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                  </apply>
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                    <real>400</real>
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                    <real>400</real>
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">u</id>
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              <real xmlns="http://schemas.mathsoft.com/math50">1.5</real>
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The integral equals π/2 when m=0, and approaches 12 as m approaches 1. The horizontal marker shows the value of Elliptick(l/10), or:

EllipticF: The Incomplete Elliptic Integral of the First Kind

1. Show the definition of the incomplete elliptic integral of the first kind, EllipticF(x, m).

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    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="*" xml:space="preserve">EllipticF</id>
        <boundVars>
          <id labels="*" xml:space="preserve">x</id>
          <id labels="*" xml:space="preserve">m</id>
        </boundVars>
      </function>
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">τ</id>
          </boundVars>
          <apply>
            <div />
            <real>1</real>
            <apply>
              <mult />
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <minus />
                  <real>1</real>
                  <apply>
                    <pow />
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                    <real>2</real>
                  </apply>
                </apply>
              </apply>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <minus />
                  <real>1</real>
                  <apply>
                    <mult />
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m</id>
                    <apply>
                      <pow />
                      <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                      <real>2</real>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <apply>
            <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">sin</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">x</id>
          </apply>
        </upperBound>
      </apply>
    </define>
  </math>
</region>

2. Evaluate EllipticF numerically.

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            <div />
            <id labels="CONSTANT" xml:space="preserve" label-is-contextual="true">π</id>
            <real>2</real>
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            <real>1</real>
            <real>5</real>
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</region>

3. Show the relationship between EllipticF and EllipticK.

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            <div />
            <id labels="CONSTANT" xml:space="preserve" label-is-contextual="true">π</id>
            <real>2</real>
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      <command>
        <placeholder />
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      <symResult>
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          <integral />
          <lambda>
            <boundVars>
              <id labels="VARIABLE" xml:space="preserve">τ</id>
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              <div />
              <real>1</real>
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                <apply>
                  <nthRoot />
                  <placeholder />
                  <apply>
                    <plus />
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                        <apply>
                          <mult />
                          <apply>
                            <pow />
                            <id labels="VARIABLE" xml:space="preserve">τ</id>
                            <real>2</real>
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                          <id labels="*" xml:space="preserve">m</id>
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                    <real>1</real>
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                </apply>
                <apply>
                  <nthRoot />
                  <placeholder />
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                    <plus />
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                      <neg />
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                        <id labels="VARIABLE" xml:space="preserve">τ</id>
                        <real>2</real>
                      </apply>
                    </apply>
                    <real>1</real>
                  </apply>
                </apply>
              </apply>
            </apply>
          </lambda>
          <lowerBound>
            <real>0</real>
          </lowerBound>
          <upperBound>
            <real>1</real>
          </upperBound>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

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    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">EllipticK</id>
        <id xml:space="preserve">m</id>
      </apply>
      <command>
        <placeholder />
      </command>
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        <apply>
          <integral />
          <lambda>
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              <id labels="VARIABLE" xml:space="preserve">τ</id>
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              <real>1</real>
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                    <plus />
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                      <neg />
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                          <apply>
                            <pow />
                            <id labels="VARIABLE" xml:space="preserve">τ</id>
                            <real>2</real>
                          </apply>
                          <id labels="*" xml:space="preserve">m</id>
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                    <real>1</real>
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                    <plus />
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                        <id labels="VARIABLE" xml:space="preserve">τ</id>
                        <real>2</real>
                      </apply>
                    </apply>
                    <real>1</real>
                  </apply>
                </apply>
              </apply>
            </apply>
          </lambda>
          <lowerBound>
            <real>0</real>
          </lowerBound>
          <upperBound>
            <real>1</real>
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        </apply>
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The two integrals are equal.

EllipticE: The Elliptic Integral of the Second Kind

1. Show the definition of the complete elliptic integral of the second kind, EllipticE(m):

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    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="VARIABLE" xml:space="preserve">EllipticE</id>
        <boundVars>
          <id labels="VARIABLE" xml:space="preserve">m</id>
        </boundVars>
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        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">τ</id>
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            <div />
            <apply>
              <nthRoot />
              <placeholder />
              <apply>
                <minus />
                <real>1</real>
                <apply>
                  <mult />
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m</id>
                  <apply>
                    <pow />
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                    <real>2</real>
                  </apply>
                </apply>
              </apply>
            </apply>
            <apply>
              <nthRoot />
              <placeholder />
              <apply>
                <minus />
                <real>1</real>
                <apply>
                  <pow />
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                  <real>2</real>
                </apply>
              </apply>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <real>1</real>
        </upperBound>
      </apply>
    </define>
  </math>
</region>

Alternatively, the function is given by:

2. Evaluate EllipticE numerically.

3. Show the definition of the incomplete elliptic integral of the second kind, EllipticE(x, m):

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  <math resultRef="58">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="VARIABLE" xml:space="preserve">EllipticEi</id>
        <boundVars>
          <id labels="VARIABLE" xml:space="preserve">x</id>
          <id labels="VARIABLE" xml:space="preserve">m</id>
        </boundVars>
      </function>
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">τ</id>
          </boundVars>
          <apply>
            <div />
            <apply>
              <nthRoot />
              <placeholder />
              <apply>
                <minus />
                <real>1</real>
                <apply>
                  <mult />
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m</id>
                  <apply>
                    <pow />
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                    <real>2</real>
                  </apply>
                </apply>
              </apply>
            </apply>
            <apply>
              <nthRoot />
              <placeholder />
              <apply>
                <minus />
                <real>1</real>
                <apply>
                  <pow />
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                  <real>2</real>
                </apply>
              </apply>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <apply>
            <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">sin</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">x</id>
          </apply>
        </upperBound>
      </apply>
    </define>
  </math>
</region>

4. Evaluate EllipticEi numerically.

5. Show the relationship between EllipticE and EllipticEi.

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    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">EllipticE</id>
        <id xml:space="preserve">m</id>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <integral />
          <lambda>
            <boundVars>
              <id labels="VARIABLE" xml:space="preserve">τ</id>
            </boundVars>
            <apply>
              <mult />
              <apply>
                <pow />
                <parens>
                  <apply>
                    <plus />
                    <apply>
                      <neg />
                      <apply>
                        <pow />
                        <id labels="VARIABLE" xml:space="preserve">τ</id>
                        <real>2</real>
                      </apply>
                    </apply>
                    <real>1</real>
                  </apply>
                </parens>
                <apply>
                  <neg />
                  <apply>
                    <div />
                    <real>1</real>
                    <real>2</real>
                  </apply>
                </apply>
              </apply>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <plus />
                  <apply>
                    <neg />
                    <parens>
                      <apply>
                        <mult />
                        <apply>
                          <pow />
                          <id labels="VARIABLE" xml:space="preserve">τ</id>
                          <real>2</real>
                        </apply>
                        <id labels="*" xml:space="preserve">m</id>
                      </apply>
                    </parens>
                  </apply>
                  <real>1</real>
                </apply>
              </apply>
            </apply>
          </lambda>
          <lowerBound>
            <real>0</real>
          </lowerBound>
          <upperBound>
            <real>1</real>
          </upperBound>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

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  <math resultRef="61">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">EllipticEi</id>
        <sequence>
          <apply>
            <div />
            <id labels="CONSTANT" xml:space="preserve" label-is-contextual="true">π</id>
            <real>2</real>
          </apply>
          <id xml:space="preserve">m</id>
        </sequence>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <integral />
          <lambda>
            <boundVars>
              <id labels="VARIABLE" xml:space="preserve">τ</id>
            </boundVars>
            <apply>
              <mult />
              <apply>
                <pow />
                <parens>
                  <apply>
                    <plus />
                    <apply>
                      <neg />
                      <apply>
                        <pow />
                        <id labels="VARIABLE" xml:space="preserve">τ</id>
                        <real>2</real>
                      </apply>
                    </apply>
                    <real>1</real>
                  </apply>
                </parens>
                <apply>
                  <neg />
                  <apply>
                    <div />
                    <real>1</real>
                    <real>2</real>
                  </apply>
                </apply>
              </apply>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <plus />
                  <apply>
                    <neg />
                    <parens>
                      <apply>
                        <mult />
                        <apply>
                          <pow />
                          <id labels="VARIABLE" xml:space="preserve">τ</id>
                          <real>2</real>
                        </apply>
                        <id labels="*" xml:space="preserve">m</id>
                      </apply>
                    </parens>
                  </apply>
                  <real>1</real>
                </apply>
              </apply>
            </apply>
          </lambda>
          <lowerBound>
            <real>0</real>
          </lowerBound>
          <upperBound>
            <real>1</real>
          </upperBound>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

The two integrals are equal.

EllipticP: The Elliptic Integral of the Third Kind

1. Show the definition of the complete elliptic integral of the third kind, EllipticPi(n, m):

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  <math resultRef="62">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="VARIABLE" xml:space="preserve">EllipticP</id>
        <boundVars>
          <id labels="VARIABLE" xml:space="preserve">n</id>
          <id labels="VARIABLE" xml:space="preserve">m</id>
        </boundVars>
      </function>
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">τ</id>
          </boundVars>
          <apply>
            <div />
            <real>1</real>
            <apply>
              <mult />
              <apply>
                <mult />
                <parens>
                  <apply>
                    <minus />
                    <real>1</real>
                    <apply>
                      <mult />
                      <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">n</id>
                      <apply>
                        <pow />
                        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                        <real>2</real>
                      </apply>
                    </apply>
                  </apply>
                </parens>
                <apply>
                  <nthRoot />
                  <placeholder />
                  <apply>
                    <minus />
                    <real>1</real>
                    <apply>
                      <pow />
                      <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                      <real>2</real>
                    </apply>
                  </apply>
                </apply>
              </apply>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <minus />
                  <real>1</real>
                  <apply>
                    <mult />
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m</id>
                    <apply>
                      <pow />
                      <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                      <real>2</real>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <real>1</real>
        </upperBound>
      </apply>
    </define>
  </math>
</region>

2. Evaluate EllipticP(n, m) numerically.

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  <math resultRef="63">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">EllipticP</id>
        <sequence>
          <apply>
            <div />
            <id labels="CONSTANT" xml:space="preserve">π</id>
            <real>4</real>
          </apply>
          <apply>
            <div />
            <real>1</real>
            <real>5</real>
          </apply>
        </sequence>
      </apply>
      <unitOverride>
        <placeholder />
      </unitOverride>
    </eval>
    <resultFormat>
      <general precision="9" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
      <matrix size="1,1" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

3. Show the definition of the incomplete elliptic integral of the third kind, EllipticPi(x, n, m):

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  <math resultRef="64">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="VARIABLE" xml:space="preserve">EllipticPi</id>
        <boundVars>
          <id labels="VARIABLE" xml:space="preserve">x</id>
          <id labels="VARIABLE" xml:space="preserve">n</id>
          <id labels="VARIABLE" xml:space="preserve">m</id>
        </boundVars>
      </function>
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">τ</id>
          </boundVars>
          <apply>
            <div />
            <real>1</real>
            <apply>
              <mult />
              <apply>
                <mult />
                <parens>
                  <apply>
                    <minus />
                    <real>1</real>
                    <apply>
                      <mult />
                      <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">n</id>
                      <apply>
                        <pow />
                        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                        <real>2</real>
                      </apply>
                    </apply>
                  </apply>
                </parens>
                <apply>
                  <nthRoot />
                  <placeholder />
                  <apply>
                    <minus />
                    <real>1</real>
                    <apply>
                      <pow />
                      <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                      <real>2</real>
                    </apply>
                  </apply>
                </apply>
              </apply>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <minus />
                  <real>1</real>
                  <apply>
                    <mult />
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m</id>
                    <apply>
                      <pow />
                      <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id>
                      <real>2</real>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <apply>
            <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">sin</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">x</id>
          </apply>
        </upperBound>
      </apply>
    </define>
  </math>
</region>

4. Evaluate EllipticPi numerically.

5. Show the relationship between EllipticP and EllipticPi.

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  <math resultRef="66">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">EllipticP</id>
        <sequence>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">n</id>
          <id xml:space="preserve">m</id>
        </sequence>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <integral />
          <lambda>
            <boundVars>
              <id labels="VARIABLE" xml:space="preserve">τ</id>
            </boundVars>
            <apply>
              <neg />
              <apply>
                <div />
                <real>1</real>
                <apply>
                  <mult />
                  <apply>
                    <mult />
                    <parens>
                      <apply>
                        <minus />
                        <apply>
                          <mult />
                          <apply>
                            <pow />
                            <id labels="VARIABLE" xml:space="preserve">τ</id>
                            <real>2</real>
                          </apply>
                          <id labels="*" xml:space="preserve">n</id>
                        </apply>
                        <real>1</real>
                      </apply>
                    </parens>
                    <apply>
                      <nthRoot />
                      <placeholder />
                      <apply>
                        <plus />
                        <apply>
                          <neg />
                          <parens>
                            <apply>
                              <mult />
                              <apply>
                                <pow />
                                <id labels="VARIABLE" xml:space="preserve">τ</id>
                                <real>2</real>
                              </apply>
                              <id labels="*" xml:space="preserve">m</id>
                            </apply>
                          </parens>
                        </apply>
                        <real>1</real>
                      </apply>
                    </apply>
                  </apply>
                  <apply>
                    <nthRoot />
                    <placeholder />
                    <apply>
                      <plus />
                      <apply>
                        <neg />
                        <apply>
                          <pow />
                          <id labels="VARIABLE" xml:space="preserve">τ</id>
                          <real>2</real>
                        </apply>
                      </apply>
                      <real>1</real>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
          </lambda>
          <lowerBound>
            <real>0</real>
          </lowerBound>
          <upperBound>
            <real>1</real>
          </upperBound>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

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  <math resultRef="67">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">EllipticPi</id>
        <sequence>
          <id xml:space="preserve">x</id>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">n</id>
          <id xml:space="preserve">m</id>
        </sequence>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <integral />
          <lambda>
            <boundVars>
              <id labels="VARIABLE" xml:space="preserve">τ</id>
            </boundVars>
            <apply>
              <neg />
              <apply>
                <div />
                <real>1</real>
                <apply>
                  <mult />
                  <apply>
                    <mult />
                    <parens>
                      <apply>
                        <minus />
                        <apply>
                          <mult />
                          <apply>
                            <pow />
                            <id labels="VARIABLE" xml:space="preserve">τ</id>
                            <real>2</real>
                          </apply>
                          <id labels="*" xml:space="preserve">n</id>
                        </apply>
                        <real>1</real>
                      </apply>
                    </parens>
                    <apply>
                      <nthRoot />
                      <placeholder />
                      <apply>
                        <plus />
                        <apply>
                          <neg />
                          <parens>
                            <apply>
                              <mult />
                              <apply>
                                <pow />
                                <id labels="VARIABLE" xml:space="preserve">τ</id>
                                <real>2</real>
                              </apply>
                              <id labels="*" xml:space="preserve">m</id>
                            </apply>
                          </parens>
                        </apply>
                        <real>1</real>
                      </apply>
                    </apply>
                  </apply>
                  <apply>
                    <nthRoot />
                    <placeholder />
                    <apply>
                      <plus />
                      <apply>
                        <neg />
                        <apply>
                          <pow />
                          <id labels="VARIABLE" xml:space="preserve">τ</id>
                          <real>2</real>
                        </apply>
                      </apply>
                      <real>1</real>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
          </lambda>
          <lowerBound>
            <real>0</real>
          </lowerBound>
          <upperBound>
            <apply>
              <id labels="FUNCTION" xml:space="preserve">sin</id>
              <id labels="*" xml:space="preserve">x</id>
            </apply>
          </upperBound>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

The two integrals are equal at x=π/2.