Ejemplo: función integral elíptica

Funciones > Funciones simbólicas > Ejemplo: función integral elíptica

Cálculo de las derivadas de las funciones integrales elípticas

• Integral elíptica completa de primera especie:

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            <id xml:space="preserve" labels="FUNCTION">ellipticK</id>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
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        </degree>
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      <command>
        <placeholder />
      </command>
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        <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>
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</region>

• Segunda derivada de la integral elíptica completa de primera especie:

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            <id xml:space="preserve">m</id>
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            <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>
        <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>
    <resultFormat>
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• Primera derivada por m de la integral elíptica incompleta de primera especie:

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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>
          <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>
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• Primera derivada por z de la integral elíptica incompleta de primera especie:

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      <command>
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      <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>
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              <nthRoot />
              <placeholder />
              <apply>
                <plus />
                <apply>
                  <neg />
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="VARIABLE">z</id>
                    <real>2</real>
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                <real>1</real>
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• Segunda derivada por m de la integral elíptica incompleta de primera especie:

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          <real>2</real>
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          <minus />
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            <apply>
              <plus />
              <apply>
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                  <apply>
                    <plus />
                    <apply>
                      <minus />
                      <apply>
                        <mult />
                        <real>3</real>
                        <apply>
                          <pow />
                          <id xml:space="preserve" labels="VARIABLE">m</id>
                          <real>2</real>
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                      </apply>
                      <apply>
                        <mult />
                        <real>5</real>
                        <id xml:space="preserve" labels="VARIABLE">m</id>
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                    <real>2</real>
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                    <id xml:space="preserve" labels="VARIABLE">z</id>
                    <id xml:space="preserve" labels="VARIABLE">m</id>
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              </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>
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              </apply>
              <apply>
                <mult />
                <real>4</real>
                <apply>
                  <pow />
                  <id xml:space="preserve" labels="VARIABLE">m</id>
                  <real>2</real>
                </apply>
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          </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>
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                </apply>
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              <apply>
                <nthRoot />
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                  <plus />
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                    <neg />
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                        <id xml:space="preserve" labels="VARIABLE">m</id>
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                          <pow />
                          <id xml:space="preserve" labels="VARIABLE">z</id>
                          <real>2</real>
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                  <real>1</real>
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• Primeras derivadas por m y z de la integral elíptica incompleta de segunda especie:

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              <real>1</real>
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              <real>1</real>
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• Primeras derivadas por m y n z de la integral elíptica incompleta de tercera especie:

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Evaluación numérica de las funciones integrales elípticas

• Utilice la palabra clave assume.

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También puede definir los argumentos de las funciones.

Calcule el límite de ellipticF cuando m se aproxima a 0:

• Utilice la palabra clave float.

• ellipticK tiene un punto de singularidad en m=1. Para m>1, los resultados serán complejos:

• Calcule la función ellipticK de un vector.

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Temas relacionados

Funciones integrales elípticas

Acerca de las funciones simbólicas

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