Exemple : Fonctions de Hankel du premier et deuxième type

Fonctions > Fonctions spéciales > Fonctions de Bessel > Exemple : Fonctions de Hankel du premier et deuxième type

Affichez la relation entre les fonctions Hankel H1 et H2 avec les fonctions Bessel Jn et Yn. Montrez également les relations entre ces fonctions et leurs versions mises à l'échelle.

Les fonctions Hankel ne sont pas définies sur 0.

1. Utilisez l'évaluation symbolique pour afficher la relation entre H1, Jn et Yn.

2. Utilisez l'évaluation symbolique pour afficher la relation entre H2, Jn et Yn.

◦ Les valeurs et les signes des composants réels de H1 et H2 sont identiques.

◦ Les valeurs des composants imaginaires de H1 et H2 sont identiques, mais leurs signes sont opposés.

3. Prenez en compte ces deux autres relations importantes entre H1 et H2.

4. Créez un tracé pour afficher visuellement |H1|=|H2|.

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        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="20">
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            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
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          <axisLabel />
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            <plotEquation>
              <math>
                <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">x</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
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        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="2">
          <axisLine position="origin" positionticmark="0" legendWidth="125.873333333333" />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <absval />
                  <apply>
                    <id labels="FUNCTION" xml:space="preserve">H1</id>
                    <sequence>
                      <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">m</id>
                      <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x</id>
                    </sequence>
                  </apply>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <absval />
                  <apply>
                    <id labels="FUNCTION" label-is-contextual="true" xml:space="preserve">H2</id>
                    <sequence>
                      <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">m</id>
                      <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x</id>
                    </sequence>
                  </apply>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="35">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <secondTickValue resultRef="36">
              <real xmlns="http://schemas.mathsoft.com/math50">0.5</real>
            </secondTickValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

5. Créez un tracé pour afficher la relation entre H1, Jn et Yn.

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        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="20">
          <axisLine position="origin" positionticmark="2" legendWidth="60.93" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math>
                <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">x</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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          <axisLine position="origin" positionticmark="0" legendWidth="125.873333333333" />
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            <gridFrequency>7</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <absval />
                  <apply>
                    <id labels="FUNCTION" xml:space="preserve">H1</id>
                    <sequence>
                      <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">m</id>
                      <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x</id>
                    </sequence>
                  </apply>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">Jn</id>
                  <sequence>
                    <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">m</id>
                    <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x</id>
                  </sequence>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" label-is-contextual="true" xml:space="preserve">Yn</id>
                  <sequence>
                    <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">m</id>
                    <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x</id>
                  </sequence>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="40">
              <real xmlns="http://schemas.mathsoft.com/math50">-1</real>
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              <real xmlns="http://schemas.mathsoft.com/math50">-0.5</real>
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              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
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  </plot>
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6. Utilisez l'évaluation symbolique pour afficher la relation entre les fonctions Hankel et leur version mise à l'échelle :

7. Créez un tracé pour montrer que :

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          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math>
                <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">x</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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          <axisLine position="origin" positionticmark="0" legendWidth="178.54" />
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            <gridFrequency>6</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <absval />
                  <apply>
                    <mult />
                    <apply>
                      <id labels="FUNCTION" xml:space="preserve">H1</id>
                      <sequence>
                        <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">m</id>
                        <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x</id>
                      </sequence>
                    </apply>
                    <apply>
                      <pow />
                      <id labels="CONSTANT" label-is-contextual="true" xml:space="preserve">e</id>
                      <apply>
                        <mult />
                        <imag symbol="i">-1.0</imag>
                        <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x</id>
                      </apply>
                    </apply>
                  </apply>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <absval />
                  <apply>
                    <id labels="FUNCTION" label-is-contextual="true" xml:space="preserve">H1.sc</id>
                    <sequence>
                      <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">m</id>
                      <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x</id>
                    </sequence>
                  </apply>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
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          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

8. Créez un tracé pour montrer que :

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      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="20">
          <axisLine position="origin" positionticmark="1" legendWidth="60.93" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math>
                <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">x</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-0.5" end="2">
          <axisLine position="origin" positionticmark="0" legendWidth="171.7" />
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            <gridFrequency>6</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <absval />
                  <apply>
                    <mult />
                    <apply>
                      <id labels="FUNCTION" xml:space="preserve">H2</id>
                      <sequence>
                        <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">m</id>
                        <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x</id>
                      </sequence>
                    </apply>
                    <apply>
                      <pow />
                      <id labels="CONSTANT" label-is-contextual="true" xml:space="preserve">e</id>
                      <apply>
                        <mult />
                        <imag symbol="i">1.0</imag>
                        <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x</id>
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                    </apply>
                  </apply>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <absval />
                  <apply>
                    <id labels="FUNCTION" label-is-contextual="true" xml:space="preserve">H2.sc</id>
                    <sequence>
                      <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">m</id>
                      <id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x</id>
                    </sequence>
                  </apply>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
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Rubriques associées

Fonctions de Hankel

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