例: 楕円積分関数

楕円積分関数の導関数の計算

• 第 1 種完全楕円積分:

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      <apply>
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        <lambda>
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            <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>
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        </lambda>
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        </degree>
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        <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>
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• 第 1 種完全楕円積分の 2 次導関数:

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    <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>
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      <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>
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      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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• 第 1 種不完全楕円積分の m による 1 次導関数:

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          <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>
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      <command>
        <placeholder />
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      <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>
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• 第 1 種不完全楕円積分の z による 1 次導関数:

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              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">z</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
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          </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>
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• 第 1 種不完全楕円積分の m による 2 次導関数:

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            <id xml:space="preserve">m</id>
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          <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>
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        </lambda>
        <degree>
          <real>2</real>
        </degree>
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          <minus />
          <apply>
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            <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>
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                  <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>
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                </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>
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                  <real>1</real>
                </apply>
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          </apply>
        </apply>
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• 第 2 種不完全楕円積分の m および z による 1 次導関数:

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                  <id xml:space="preserve" labels="VARIABLE">z</id>
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              <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
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                <id xml:space="preserve" labels="VARIABLE">m</id>
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          <apply>
            <mult />
            <real>2</real>
            <id xml:space="preserve" labels="VARIABLE">m</id>
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              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">z</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
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            <apply>
              <plus />
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                <neg />
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                    <mult />
                    <id xml:space="preserve" labels="VARIABLE">m</id>
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">z</id>
                      <real>2</real>
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              <real>1</real>
            </apply>
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          <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>
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• 第 3 種不完全楕円積分の m、n、および z による 1 次導関数:

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            <id xml:space="preserve">m</id>
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          <apply>
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                  </apply>
                  <real>1</real>
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              </apply>
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          </apply>
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    <resultFormat>
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楕円積分関数の数値的な評価

• assume キーワードを使用します。

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            <id xml:space="preserve">m</id>
            <real>0</real>
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          <real>2</real>
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関数の引数を設定することもできます。

m が 0 に近づくときに ellipticF の限界を計算します。

• float キーワードを使用します。

• ellipticK は m=1 において特異点を持ちます。m>1 の場合、結果は複素数になります。

• ベクトルの ellipticK を計算します。

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          <real>1.5</real>
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          <real>1.415737208425956295</real>
          <real>1.8540746773013719184</real>
          <id xml:space="preserve" labels="CONSTANT">∞</id>
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            <minus />
            <real>1.6566381702365939876</real>
            <imag symbol="i">1.4157372084259560729</imag>
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