Example: Elliptic Integral Function

Functions > Symbolic Functions > Example: Elliptic Integral Function

Calculating the Derivatives of Elliptic Integral Functions

• The complete elliptic integral of the first kind:

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  <math resultRef="16">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <derivative />
        <lambda>
          <boundVars>
            <id xml:space="preserve">m</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION">ellipticK</id>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
          </apply>
        </lambda>
        <degree>
          <placeholder />
        </degree>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <div />
          <apply>
            <plus />
            <apply>
              <neg />
              <apply>
                <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
                <id xml:space="preserve" labels="VARIABLE">m</id>
              </apply>
            </apply>
            <apply>
              <mult />
              <parens>
                <apply>
                  <minus />
                  <real>1</real>
                  <id xml:space="preserve" labels="VARIABLE">m</id>
                </apply>
              </parens>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">ellipticK</id>
                <id xml:space="preserve" labels="VARIABLE">m</id>
              </apply>
            </apply>
          </apply>
          <apply>
            <minus />
            <apply>
              <mult />
              <real>2</real>
              <apply>
                <pow />
                <id xml:space="preserve" labels="VARIABLE">m</id>
                <real>2</real>
              </apply>
            </apply>
            <apply>
              <mult />
              <real>2</real>
              <id xml:space="preserve" labels="VARIABLE">m</id>
            </apply>
          </apply>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

• Second derivative of the complete elliptic integral of the first kind:

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  <math resultRef="17">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <derivative />
        <lambda>
          <boundVars>
            <id xml:space="preserve">m</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION">ellipticK</id>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
          </apply>
        </lambda>
        <degree>
          <real>2</real>
        </degree>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <div />
          <apply>
            <plus />
            <apply>
              <mult />
              <parens>
                <apply>
                  <minus />
                  <apply>
                    <mult />
                    <real>4</real>
                    <id xml:space="preserve" labels="VARIABLE">m</id>
                  </apply>
                  <real>2</real>
                </apply>
              </parens>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
                <id xml:space="preserve" labels="VARIABLE">m</id>
              </apply>
            </apply>
            <apply>
              <mult />
              <parens>
                <apply>
                  <plus />
                  <apply>
                    <minus />
                    <apply>
                      <mult />
                      <real>3</real>
                      <apply>
                        <pow />
                        <id xml:space="preserve" labels="VARIABLE">m</id>
                        <real>2</real>
                      </apply>
                    </apply>
                    <apply>
                      <mult />
                      <real>5</real>
                      <id xml:space="preserve" labels="VARIABLE">m</id>
                    </apply>
                  </apply>
                  <real>2</real>
                </apply>
              </parens>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">ellipticK</id>
                <id xml:space="preserve" labels="VARIABLE">m</id>
              </apply>
            </apply>
          </apply>
          <apply>
            <plus />
            <apply>
              <minus />
              <apply>
                <mult />
                <real>4</real>
                <apply>
                  <pow />
                  <id xml:space="preserve" labels="VARIABLE">m</id>
                  <real>4</real>
                </apply>
              </apply>
              <apply>
                <mult />
                <real>8</real>
                <apply>
                  <pow />
                  <id xml:space="preserve" labels="VARIABLE">m</id>
                  <real>3</real>
                </apply>
              </apply>
            </apply>
            <apply>
              <mult />
              <real>4</real>
              <apply>
                <pow />
                <id xml:space="preserve" labels="VARIABLE">m</id>
                <real>2</real>
              </apply>
            </apply>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• First derivative by m of the incomplete elliptic integral of the first kind:

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  <math resultRef="18">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <derivative />
        <lambda>
          <boundVars>
            <id xml:space="preserve">m</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION">ellipticF</id>
            <sequence>
              <id xml:space="preserve">z</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
            </sequence>
          </apply>
        </lambda>
        <degree>
          <placeholder />
        </degree>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <plus />
          <apply>
            <neg />
            <apply>
              <div />
              <apply>
                <plus />
                <apply>
                  <mult />
                  <parens>
                    <apply>
                      <minus />
                      <id xml:space="preserve" labels="VARIABLE">m</id>
                      <real>1</real>
                    </apply>
                  </parens>
                  <apply>
                    <id xml:space="preserve" labels="FUNCTION">ellipticF</id>
                    <sequence>
                      <id xml:space="preserve" labels="VARIABLE">z</id>
                      <id xml:space="preserve" labels="VARIABLE">m</id>
                    </sequence>
                  </apply>
                </apply>
                <apply>
                  <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
                  <sequence>
                    <id xml:space="preserve" labels="VARIABLE">z</id>
                    <id xml:space="preserve" labels="VARIABLE">m</id>
                  </sequence>
                </apply>
              </apply>
              <apply>
                <minus />
                <apply>
                  <mult />
                  <real>2</real>
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="VARIABLE">m</id>
                    <real>2</real>
                  </apply>
                </apply>
                <apply>
                  <mult />
                  <real>2</real>
                  <id xml:space="preserve" labels="VARIABLE">m</id>
                </apply>
              </apply>
            </apply>
          </apply>
          <apply>
            <div />
            <apply>
              <mult />
              <id xml:space="preserve" labels="VARIABLE">z</id>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <plus />
                  <apply>
                    <neg />
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">z</id>
                      <real>2</real>
                    </apply>
                  </apply>
                  <real>1</real>
                </apply>
              </apply>
            </apply>
            <apply>
              <mult />
              <parens>
                <apply>
                  <minus />
                  <apply>
                    <mult />
                    <real>2</real>
                    <id xml:space="preserve" labels="VARIABLE">m</id>
                  </apply>
                  <real>2</real>
                </apply>
              </parens>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <plus />
                  <apply>
                    <neg />
                    <parens>
                      <apply>
                        <mult />
                        <id xml:space="preserve" labels="VARIABLE">m</id>
                        <apply>
                          <pow />
                          <id xml:space="preserve" labels="VARIABLE">z</id>
                          <real>2</real>
                        </apply>
                      </apply>
                    </parens>
                  </apply>
                  <real>1</real>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• First derivative by z of the incomplete elliptic integral of the first kind:

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  <math resultRef="19">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <derivative />
        <lambda>
          <boundVars>
            <id xml:space="preserve">z</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION">ellipticF</id>
            <sequence>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">z</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
            </sequence>
          </apply>
        </lambda>
        <degree>
          <placeholder />
        </degree>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <div />
          <real>1</real>
          <apply>
            <mult />
            <apply>
              <nthRoot />
              <placeholder />
              <apply>
                <plus />
                <apply>
                  <neg />
                  <parens>
                    <apply>
                      <mult />
                      <id xml:space="preserve" labels="VARIABLE">m</id>
                      <apply>
                        <pow />
                        <id xml:space="preserve" labels="VARIABLE">z</id>
                        <real>2</real>
                      </apply>
                    </apply>
                  </parens>
                </apply>
                <real>1</real>
              </apply>
            </apply>
            <apply>
              <nthRoot />
              <placeholder />
              <apply>
                <plus />
                <apply>
                  <neg />
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="VARIABLE">z</id>
                    <real>2</real>
                  </apply>
                </apply>
                <real>1</real>
              </apply>
            </apply>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• Second derivative by m of the incomplete elliptic integral of the first kind:

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  <math resultRef="20">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <derivative />
        <lambda>
          <boundVars>
            <id xml:space="preserve">m</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION">ellipticF</id>
            <sequence>
              <id xml:space="preserve">z</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
            </sequence>
          </apply>
        </lambda>
        <degree>
          <real>2</real>
        </degree>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <minus />
          <apply>
            <div />
            <apply>
              <plus />
              <apply>
                <mult />
                <parens>
                  <apply>
                    <plus />
                    <apply>
                      <minus />
                      <apply>
                        <mult />
                        <real>3</real>
                        <apply>
                          <pow />
                          <id xml:space="preserve" labels="VARIABLE">m</id>
                          <real>2</real>
                        </apply>
                      </apply>
                      <apply>
                        <mult />
                        <real>5</real>
                        <id xml:space="preserve" labels="VARIABLE">m</id>
                      </apply>
                    </apply>
                    <real>2</real>
                  </apply>
                </parens>
                <apply>
                  <id xml:space="preserve" labels="FUNCTION">ellipticF</id>
                  <sequence>
                    <id xml:space="preserve" labels="VARIABLE">z</id>
                    <id xml:space="preserve" labels="VARIABLE">m</id>
                  </sequence>
                </apply>
              </apply>
              <apply>
                <mult />
                <parens>
                  <apply>
                    <minus />
                    <apply>
                      <mult />
                      <real>4</real>
                      <id xml:space="preserve" labels="VARIABLE">m</id>
                    </apply>
                    <real>2</real>
                  </apply>
                </parens>
                <apply>
                  <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
                  <sequence>
                    <id xml:space="preserve" labels="VARIABLE">z</id>
                    <id xml:space="preserve" labels="VARIABLE">m</id>
                  </sequence>
                </apply>
              </apply>
            </apply>
            <apply>
              <plus />
              <apply>
                <minus />
                <apply>
                  <mult />
                  <real>4</real>
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="VARIABLE">m</id>
                    <real>4</real>
                  </apply>
                </apply>
                <apply>
                  <mult />
                  <real>8</real>
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="VARIABLE">m</id>
                    <real>3</real>
                  </apply>
                </apply>
              </apply>
              <apply>
                <mult />
                <real>4</real>
                <apply>
                  <pow />
                  <id xml:space="preserve" labels="VARIABLE">m</id>
                  <real>2</real>
                </apply>
              </apply>
            </apply>
          </apply>
          <apply>
            <div />
            <apply>
              <mult />
              <parens>
                <apply>
                  <plus />
                  <apply>
                    <mult />
                    <parens>
                      <apply>
                        <minus />
                        <apply>
                          <mult />
                          <real>4</real>
                          <apply>
                            <pow />
                            <id xml:space="preserve" labels="VARIABLE">m</id>
                            <real>2</real>
                          </apply>
                        </apply>
                        <apply>
                          <mult />
                          <real>2</real>
                          <id xml:space="preserve" labels="VARIABLE">m</id>
                        </apply>
                      </apply>
                    </parens>
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">z</id>
                      <real>3</real>
                    </apply>
                  </apply>
                  <apply>
                    <mult />
                    <parens>
                      <apply>
                        <minus />
                        <real>1</real>
                        <apply>
                          <mult />
                          <real>3</real>
                          <id xml:space="preserve" labels="VARIABLE">m</id>
                        </apply>
                      </apply>
                    </parens>
                    <id xml:space="preserve" labels="VARIABLE">z</id>
                  </apply>
                </apply>
              </parens>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <plus />
                  <apply>
                    <neg />
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">z</id>
                      <real>2</real>
                    </apply>
                  </apply>
                  <real>1</real>
                </apply>
              </apply>
            </apply>
            <apply>
              <mult />
              <parens>
                <apply>
                  <plus />
                  <apply>
                    <mult />
                    <parens>
                      <apply>
                        <plus />
                        <apply>
                          <minus />
                          <apply>
                            <mult />
                            <real>4</real>
                            <apply>
                              <pow />
                              <id xml:space="preserve" labels="VARIABLE">m</id>
                              <real>4</real>
                            </apply>
                          </apply>
                          <apply>
                            <mult />
                            <real>8</real>
                            <apply>
                              <pow />
                              <id xml:space="preserve" labels="VARIABLE">m</id>
                              <real>3</real>
                            </apply>
                          </apply>
                        </apply>
                        <apply>
                          <mult />
                          <real>4</real>
                          <apply>
                            <pow />
                            <id xml:space="preserve" labels="VARIABLE">m</id>
                            <real>2</real>
                          </apply>
                        </apply>
                      </apply>
                    </parens>
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">z</id>
                      <real>2</real>
                    </apply>
                  </apply>
                  <parens>
                    <apply>
                      <minus />
                      <apply>
                        <minus />
                        <apply>
                          <mult />
                          <real>8</real>
                          <apply>
                            <pow />
                            <id xml:space="preserve" labels="VARIABLE">m</id>
                            <real>2</real>
                          </apply>
                        </apply>
                        <apply>
                          <mult />
                          <real>4</real>
                          <apply>
                            <pow />
                            <id xml:space="preserve" labels="VARIABLE">m</id>
                            <real>3</real>
                          </apply>
                        </apply>
                      </apply>
                      <apply>
                        <mult />
                        <real>4</real>
                        <id xml:space="preserve" labels="VARIABLE">m</id>
                      </apply>
                    </apply>
                  </parens>
                </apply>
              </parens>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <plus />
                  <apply>
                    <neg />
                    <parens>
                      <apply>
                        <mult />
                        <id xml:space="preserve" labels="VARIABLE">m</id>
                        <apply>
                          <pow />
                          <id xml:space="preserve" labels="VARIABLE">z</id>
                          <real>2</real>
                        </apply>
                      </apply>
                    </parens>
                  </apply>
                  <real>1</real>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• First derivatives by m and z of the incomplete elliptic integral of the second kind:

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  <math resultRef="21">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <derivative />
        <lambda>
          <boundVars>
            <id xml:space="preserve">m</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
            <sequence>
              <id xml:space="preserve">z</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
            </sequence>
          </apply>
        </lambda>
        <degree>
          <placeholder />
        </degree>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <div />
          <apply>
            <plus />
            <apply>
              <neg />
              <apply>
                <id xml:space="preserve" labels="FUNCTION">ellipticF</id>
                <sequence>
                  <id xml:space="preserve" labels="VARIABLE">z</id>
                  <id xml:space="preserve" labels="VARIABLE">m</id>
                </sequence>
              </apply>
            </apply>
            <apply>
              <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
              <sequence>
                <id xml:space="preserve" labels="VARIABLE">z</id>
                <id xml:space="preserve" labels="VARIABLE">m</id>
              </sequence>
            </apply>
          </apply>
          <apply>
            <mult />
            <real>2</real>
            <id xml:space="preserve" labels="VARIABLE">m</id>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

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  <math resultRef="22">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <derivative />
        <lambda>
          <boundVars>
            <id xml:space="preserve">z</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
            <sequence>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">z</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
            </sequence>
          </apply>
        </lambda>
        <degree>
          <placeholder />
        </degree>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <div />
          <apply>
            <nthRoot />
            <placeholder />
            <apply>
              <plus />
              <apply>
                <neg />
                <parens>
                  <apply>
                    <mult />
                    <id xml:space="preserve" labels="VARIABLE">m</id>
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">z</id>
                      <real>2</real>
                    </apply>
                  </apply>
                </parens>
              </apply>
              <real>1</real>
            </apply>
          </apply>
          <apply>
            <nthRoot />
            <placeholder />
            <apply>
              <plus />
              <apply>
                <neg />
                <apply>
                  <pow />
                  <id xml:space="preserve" labels="VARIABLE">z</id>
                  <real>2</real>
                </apply>
              </apply>
              <real>1</real>
            </apply>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• First derivatives by m, n and z of the incomplete elliptic integral of the third kind:

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  <math resultRef="23">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <derivative />
        <lambda>
          <boundVars>
            <id xml:space="preserve">m</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION">ellipticPi</id>
            <sequence>
              <id xml:space="preserve">z</id>
              <id xml:space="preserve">n</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
            </sequence>
          </apply>
        </lambda>
        <degree>
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Numerically Evaluating Elliptical integral functions

• Use the assume keyword.

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You can also set the functions arguments.

Calculate the limit of ellipticF when m approaches 0:

• Use the float keyword.

• ellipticK has a singularity point at m=1. For m>1, results will be complex:

• Calculate the ellipticK of a vector.

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