Exemple : Fonction intégrale elliptique

Fonctions > Fonctions symboliques > Exemple : Fonction intégrale elliptique

Calcul des dérivées de fonctions intégrales elliptiques

• L'intégrale elliptique complète du premier type :

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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>
              <id xml:space="preserve" labels="VARIABLE">m</id>
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• Dérivée seconde de l'intégrale elliptique complète du premier type :

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            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m</id>
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        <degree>
          <real>2</real>
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        <apply>
          <div />
          <apply>
            <plus />
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                <apply>
                  <minus />
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                    <mult />
                    <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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                      <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>
              </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>
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• Dérivée première par m de l'intégrale elliptique incomplète du premier type :

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                      <real>1</real>
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                    <id xml:space="preserve" labels="VARIABLE">m</id>
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                <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>
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          </apply>
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            <apply>
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              <id xml:space="preserve" labels="VARIABLE">z</id>
              <apply>
                <nthRoot />
                <placeholder />
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                  <plus />
                  <apply>
                    <neg />
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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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                    <real>2</real>
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                  <real>2</real>
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                          <id xml:space="preserve" labels="VARIABLE">z</id>
                          <real>2</real>
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                  <real>1</real>
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• Dérivée première par z de l'intégrale elliptique incomplète du premier type :

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                      <id xml:space="preserve" labels="VARIABLE">m</id>
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• Dérivée seconde par m de l'intégrale elliptique incomplète du premier type :

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                          <real>4</real>
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                            <pow />
                            <id xml:space="preserve" labels="VARIABLE">m</id>
                            <real>2</real>
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                      <real>3</real>
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                        <real>1</real>
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                          <real>3</real>
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                    <neg />
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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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            <apply>
              <mult />
              <parens>
                <apply>
                  <plus />
                  <apply>
                    <mult />
                    <parens>
                      <apply>
                        <plus />
                        <apply>
                          <minus />
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                            <mult />
                            <real>4</real>
                            <apply>
                              <pow />
                              <id xml:space="preserve" labels="VARIABLE">m</id>
                              <real>4</real>
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                          <apply>
                            <mult />
                            <real>8</real>
                            <apply>
                              <pow />
                              <id xml:space="preserve" labels="VARIABLE">m</id>
                              <real>3</real>
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                          </apply>
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                        <apply>
                          <mult />
                          <real>4</real>
                          <apply>
                            <pow />
                            <id xml:space="preserve" labels="VARIABLE">m</id>
                            <real>2</real>
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                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">z</id>
                      <real>2</real>
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                      <minus />
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                        <minus />
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                          <mult />
                          <real>8</real>
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                            <pow />
                            <id xml:space="preserve" labels="VARIABLE">m</id>
                            <real>2</real>
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                        <apply>
                          <mult />
                          <real>4</real>
                          <apply>
                            <pow />
                            <id xml:space="preserve" labels="VARIABLE">m</id>
                            <real>3</real>
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                      <apply>
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                        <real>4</real>
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• Dérivées premières par m et z de l'intégrale elliptique incomplète du second type :

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          <apply>
            <mult />
            <real>2</real>
            <id xml:space="preserve" labels="VARIABLE">m</id>
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• Dérivées premières par m, n et z de l'intégrale elliptique incomplète du troisième type :

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Evaluation numérique des fonctions intégrales elliptiques

• Utilisez le mot-clé assume.

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            <id xml:space="preserve">m</id>
            <real>0</real>
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          <real>2</real>
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Vous pouvez également définir les arguments des fonctions.

Calculez la limite de ellipticF lorsque m approche de 0 :

• Utilisez le mot-clé float.

• ellipticK possède un point de singularité à m=1. Pour m>1, les résultats seront complexes :

• Calculez le ellipticK d'un vecteur.

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          <real>1.5</real>
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          <real>1,8540746773013719184</real>
          <id xml:space="preserve" labels="CONSTANT">∞</id>
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            <imag symbol="i">1,4157372084259560729</imag>
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Rubriques associées

Fonctions intégrales elliptiques

A propos des fonctions symboliques

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