Beispiel: Elliptische Integralfunktion

Funktionen > Symbolische Funktionen > Beispiel: Elliptische Integralfunktion

Ableitungen elliptischer Integralfunktionen berechnen

• Vollständiges elliptisches Integral erster Art:

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                <id xml:space="preserve" labels="VARIABLE">m</id>
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                  <id xml:space="preserve" labels="VARIABLE">m</id>
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                <id xml:space="preserve" labels="FUNCTION">ellipticK</id>
                <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">m</id>
                <real>2</real>
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              <real>2</real>
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• Zweite Ableitung des vollständigen elliptischen Integrals erster Art:

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            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
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          <apply>
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                    <real>4</real>
                    <id xml:space="preserve" labels="VARIABLE">m</id>
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                  <real>2</real>
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              <apply>
                <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
                <id xml:space="preserve" labels="VARIABLE">m</id>
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            <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>
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                      <mult />
                      <real>5</real>
                      <id xml:space="preserve" labels="VARIABLE">m</id>
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                  <real>2</real>
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              <apply>
                <id xml:space="preserve" labels="FUNCTION">ellipticK</id>
                <id xml:space="preserve" labels="VARIABLE">m</id>
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            </apply>
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          <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>
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• Erste Ableitung durch m des unvollständigen elliptischen Integrals erster Art:

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            <id xml:space="preserve" labels="FUNCTION">ellipticF</id>
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              <id xml:space="preserve">z</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
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                      <id xml:space="preserve" labels="VARIABLE">m</id>
                      <real>1</real>
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                  <apply>
                    <id xml:space="preserve" labels="FUNCTION">ellipticF</id>
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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>
                  <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
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                    <id xml:space="preserve" labels="VARIABLE">z</id>
                    <id xml:space="preserve" labels="VARIABLE">m</id>
                  </sequence>
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              </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>
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            </apply>
          </apply>
          <apply>
            <div />
            <apply>
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              <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>
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                  <real>1</real>
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            </apply>
            <apply>
              <mult />
              <parens>
                <apply>
                  <minus />
                  <apply>
                    <mult />
                    <real>2</real>
                    <id xml:space="preserve" labels="VARIABLE">m</id>
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                  <real>2</real>
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              <apply>
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                <placeholder />
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                          <id xml:space="preserve" labels="VARIABLE">z</id>
                          <real>2</real>
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• Erste Ableitung durch z des unvollständigen elliptischen Integrals erster Art:

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      <command>
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          <real>1</real>
          <apply>
            <mult />
            <apply>
              <nthRoot />
              <placeholder />
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                <plus />
                <apply>
                  <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>
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                    <real>2</real>
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• Zweite Ableitung durch m des unvollständigen elliptischen Integrals erster Art:

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                      <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">m</id>
                    <real>4</real>
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            <apply>
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              <parens>
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                  <apply>
                    <mult />
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                      <apply>
                        <minus />
                        <apply>
                          <mult />
                          <real>4</real>
                          <apply>
                            <pow />
                            <id xml:space="preserve" labels="VARIABLE">m</id>
                            <real>2</real>
                          </apply>
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                        <apply>
                          <mult />
                          <real>2</real>
                          <id xml:space="preserve" labels="VARIABLE">m</id>
                        </apply>
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                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">z</id>
                      <real>3</real>
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                    <mult />
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                        <minus />
                        <real>1</real>
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                          <real>3</real>
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                    <id xml:space="preserve" labels="VARIABLE">z</id>
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                <nthRoot />
                <placeholder />
                <apply>
                  <plus />
                  <apply>
                    <neg />
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">z</id>
                      <real>2</real>
                    </apply>
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                  <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>
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                    </parens>
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">z</id>
                      <real>2</real>
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                      <minus />
                      <apply>
                        <minus />
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                          <mult />
                          <real>8</real>
                          <apply>
                            <pow />
                            <id xml:space="preserve" labels="VARIABLE">m</id>
                            <real>2</real>
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                          <real>4</real>
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                            <real>3</real>
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                      <apply>
                        <mult />
                        <real>4</real>
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• Erste Ableitungen durch m und z des unvollständigen elliptischen Integrals zweiter Art:

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                <id xml:space="preserve" labels="VARIABLE">m</id>
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• Erste Ableitungen durch m, n und z des unvollständigen elliptischen Integrals dritter Art:

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                        </apply>
                      </apply>
                    </parens>
                  </apply>
                  <real>1</real>
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    <resultFormat>
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                      <apply>
                        <mult />
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                        <id xml:space="preserve" labels="VARIABLE">m</id>
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                      <id xml:space="preserve" labels="VARIABLE">n</id>
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                  <pow />
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                  <real>2</real>
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                    <apply>
                      <mult />
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                            <id xml:space="preserve" labels="VARIABLE">m</id>
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                    <real>1</real>
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                  <real>1</real>
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Elliptische Integralfunktionen numerisch auswerten

• Verwenden Sie das Schlüsselwort assume.

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              <real>1</real>
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          <apply>
            <equal />
            <id xml:space="preserve">m</id>
            <real>0</real>
          </apply>
        </sequence>
      </command>
      <symResult>
        <apply>
          <div />
          <id xml:space="preserve" labels="CONSTANT">π</id>
          <real>2</real>
        </apply>
      </symResult>
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Sie können auch die Funktionsargumente festlegen.

Berechnen Sie den Grenzwert von ellipticF bei m Annäherungen an 0:

• Verwenden Sie das Schlüsselwort float.

• ellipticK hat einen Singularitätspunkt bei m=1. Für m>1 sind die Ergebnisse komplex:

• Berechnen Sie das ellipticK eines Vektors.

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          <real>-0.5</real>
          <real>0.5</real>
          <real>1</real>
          <real>1.5</real>
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      <command>
        <placeholder />
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          <real>1.415737208425956295</real>
          <real>1.8540746773013719184</real>
          <id xml:space="preserve" labels="CONSTANT">∞</id>
          <apply>
            <minus />
            <real>1.6566381702365939876</real>
            <imag symbol="i">1.4157372084259560729</imag>
          </apply>
        </matrix>
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Zugehörige Themen

Elliptische Integralfunktionen

Symbolische Funktionen

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