Beispiel: Modifizierte Bessel-Funktionen der zweiten Art

Funktionen > Spezielle Funktionen > Bessel-Funktionen > Beispiel: Modifizierte Bessel-Funktionen der zweiten Art

Zeigen Sie die Beziehung zwischen den Funktionen K0, K1 und Kn an. Zeigen Sie auch die Beziehungen zwischen diesen Funktionen und ihren skalierten Versionen an.

1. Legen Sie zwei Schrittbereichsvariablen fest:

2. Plotten Sie die Funktionen K0 und K1. Fügen Sie dem Diagramm die Funktion Kn zweiter Ordnung hinzu:

<region id="ID0ENSAAB" height="316.8" width="574.66857242139008" top="364.80000000000007" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white" origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace resultRef="219">
          <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
        <trace resultRef="220">
          <traceStyle color="#FF068149" symbol="none" line-weight="1" line-style="Solid">Linien</traceStyle>
        </trace>
        <trace resultRef="221">
          <traceStyle color="#FF2E3192" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
      </traces>
      <graph-size width="417.06857242139" height="240" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-1" end="5">
          <axisLine position="origin" positionticmark="0" legendWidth="60.93" />
          <axisGrid>
            <gridFrequency>7</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="228">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">x</id>
              </math>
              <math resultRef="229">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="223">
              <real xmlns="http://schemas.mathsoft.com/math50">-1</real>
            </startValue>
            <secondTickValue resultRef="225">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </secondTickValue>
            <endValue resultRef="227">
              <real xmlns="http://schemas.mathsoft.com/math50">5</real>
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="10">
          <axisLine position="origin" positionticmark="1" legendWidth="117.57" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="236">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">K0</id>
                  <id xml:space="preserve">x</id>
                </apply>
              </math>
              <math resultRef="237">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="238">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">K1</id>
                  <id xml:space="preserve">x</id>
                </apply>
              </math>
              <math resultRef="239">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="240">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">Kn</id>
                  <sequence>
                    <id xml:space="preserve">m</id>
                    <id xml:space="preserve">x</id>
                  </sequence>
                </apply>
              </math>
              <math resultRef="241">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="231">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <secondTickValue resultRef="233">
              <real xmlns="http://schemas.mathsoft.com/math50">1</real>
            </secondTickValue>
            <endValue resultRef="235">
              <real xmlns="http://schemas.mathsoft.com/math50">10</real>
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

3. Plotten Sie die Funktion Kn fünfter und achter Ordnung:

<region id="ID0EBUAAB" height="316.79999999999995" width="574.86857242139" top="902.40000000000009" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white" origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace resultRef="256">
          <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
        <trace resultRef="257">
          <traceStyle color="#FF068149" symbol="none" line-weight="1" line-style="Solid">Linien</traceStyle>
        </trace>
        <trace resultRef="258">
          <traceStyle color="#FF2E3192" symbol="none" line-weight="1" line-style="Solid">Linien</traceStyle>
        </trace>
        <trace resultRef="259">
          <traceStyle color="#FF2E3192" symbol="none" line-weight="1" line-style="Dash">Linien</traceStyle>
        </trace>
        <trace resultRef="260">
          <traceStyle color="#FFCA5053" symbol="none" line-weight="1" line-style="Dash">lines</traceStyle>
        </trace>
      </traces>
      <graph-size width="398.06857242139" height="240" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-1" end="7">
          <axisLine position="origin" positionticmark="0" legendWidth="60.93" />
          <axisGrid>
            <gridFrequency>9</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="267">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">x</id>
              </math>
              <math resultRef="268">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="262">
              <real xmlns="http://schemas.mathsoft.com/math50">-1</real>
            </startValue>
            <secondTickValue resultRef="264">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </secondTickValue>
            <endValue resultRef="266">
              <real xmlns="http://schemas.mathsoft.com/math50">7</real>
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="10">
          <axisLine position="origin" positionticmark="1" legendWidth="140.3" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="275">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">K0</id>
                  <id xml:space="preserve">x</id>
                </apply>
              </math>
              <math resultRef="276">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="277">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">K1</id>
                  <id xml:space="preserve">x</id>
                </apply>
              </math>
              <math resultRef="278">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="279">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">Kn</id>
                  <sequence>
                    <id xml:space="preserve">m</id>
                    <id xml:space="preserve">x</id>
                  </sequence>
                </apply>
              </math>
              <math resultRef="280">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="281">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">Kn</id>
                  <sequence>
                    <apply>
                      <plus />
                      <id xml:space="preserve">m</id>
                      <real>3</real>
                    </apply>
                    <id xml:space="preserve">x</id>
                  </sequence>
                </apply>
              </math>
              <math resultRef="282">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="283">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">Kn</id>
                  <sequence>
                    <apply>
                      <plus />
                      <id xml:space="preserve">m</id>
                      <real>6</real>
                    </apply>
                    <id xml:space="preserve">x</id>
                  </sequence>
                </apply>
              </math>
              <math resultRef="284">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="270">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <secondTickValue resultRef="272">
              <real xmlns="http://schemas.mathsoft.com/math50">1</real>
            </secondTickValue>
            <endValue resultRef="274">
              <real xmlns="http://schemas.mathsoft.com/math50">10</real>
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

• Je höher die Ordnung der Funktion Kn, desto größer die Verschiebung der Funktion nach rechts.

• Alle Ordnungen der Funktion Kn nähern sich unendlich bei x=0.

4. Plotten Sie die Funktion Kn mit einem kleinen Unterschied in m, um zu zeigen, dass sie sich der Unendlichkeit mit derselben Geschwindigkeit nähern:

<region id="ID0EDZAAB" height="316.80000000000007" width="556.86857242139013" top="1382.4" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white" origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace resultRef="303">
          <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
        <trace resultRef="304">
          <traceStyle color="#FF068149" symbol="none" line-weight="1" line-style="Solid">Linien</traceStyle>
        </trace>
        <trace resultRef="305">
          <traceStyle color="#FF2E3192" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
      </traces>
      <graph-size width="380.06857242139" height="240" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-1" end="10">
          <axisLine position="origin" positionticmark="0" legendWidth="60.93" />
          <axisGrid>
            <gridFrequency>12</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="312">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">x</id>
              </math>
              <math resultRef="313">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="307">
              <real xmlns="http://schemas.mathsoft.com/math50">-1</real>
            </startValue>
            <secondTickValue resultRef="309">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </secondTickValue>
            <endValue resultRef="311">
              <real xmlns="http://schemas.mathsoft.com/math50">10</real>
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="10">
          <axisLine position="origin" positionticmark="1" legendWidth="152.65" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="320">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">Kn</id>
                  <sequence>
                    <id xml:space="preserve">m</id>
                    <id xml:space="preserve">x</id>
                  </sequence>
                </apply>
              </math>
              <math resultRef="321">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="322">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">Kn</id>
                  <sequence>
                    <apply>
                      <plus />
                      <id xml:space="preserve">m</id>
                      <real>0.2</real>
                    </apply>
                    <id xml:space="preserve">x</id>
                  </sequence>
                </apply>
              </math>
              <math resultRef="323">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="324">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">Kn</id>
                  <sequence>
                    <apply>
                      <plus />
                      <id xml:space="preserve">m</id>
                      <real>0.4</real>
                    </apply>
                    <id xml:space="preserve">x</id>
                  </sequence>
                </apply>
              </math>
              <math resultRef="325">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="315">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <secondTickValue resultRef="317">
              <real xmlns="http://schemas.mathsoft.com/math50">1</real>
            </secondTickValue>
            <endValue resultRef="319">
              <real xmlns="http://schemas.mathsoft.com/math50">10</real>
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

5. Erstellen Sie ein Diagramm, um zu zeigen, dass K0(y)=Kn(0,y) Setzen Sie die Teilstrichwerte zurück, um die x-Achse zu vergrößern, damit mehr Details angezeigt werden:

<region id="ID0EX1AAB" height="297.6" width="576.46857242138935" top="1785.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white" origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace resultRef="340">
          <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
        <trace resultRef="341">
          <traceStyle color="#FF068149" symbol="circle" line-weight="1" line-style="None">lines</traceStyle>
        </trace>
      </traces>
      <graph-size width="438.068572421389" height="240" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="5">
          <axisLine position="origin" positionticmark="0" legendWidth="60.74" />
          <axisGrid>
            <gridFrequency>6</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="344">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">y</id>
              </math>
              <math resultRef="345">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <secondTickValue resultRef="343">
              <real xmlns="http://schemas.mathsoft.com/math50">1</real>
            </secondTickValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="2">
          <axisLine position="origin" positionticmark="0" legendWidth="111.706666666667" />
          <axisGrid>
            <gridFrequency>3</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="352">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">K0</id>
                  <id xml:space="preserve">y</id>
                </apply>
              </math>
              <math resultRef="353">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="354">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">Kn</id>
                  <sequence>
                    <real>0</real>
                    <id xml:space="preserve">y</id>
                  </sequence>
                </apply>
              </math>
              <math resultRef="355">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="347">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <secondTickValue resultRef="349">
              <real xmlns="http://schemas.mathsoft.com/math50">1</real>
            </secondTickValue>
            <endValue resultRef="351">
              <real xmlns="http://schemas.mathsoft.com/math50">2</real>
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

6. Erstellen Sie ein Diagramm, um zu zeigen, dass K1(y)=Kn(1,y) Setzen Sie die Teilstrichwerte zurück, um die x-Achse zu vergrößern, damit mehr Details angezeigt werden:

<region id="ID0EL3AAB" height="297.6" width="576.46857242139015" top="2208.0000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white" origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace resultRef="366">
          <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
        <trace resultRef="367">
          <traceStyle color="#FF2E3192" symbol="circle" line-weight="1" line-style="None">lines</traceStyle>
        </trace>
      </traces>
      <graph-size width="438.06857242139" height="240" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="5">
          <axisLine position="origin" positionticmark="0" legendWidth="60.74" />
          <axisGrid>
            <gridFrequency>6</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="370">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">y</id>
              </math>
              <math resultRef="371">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <secondTickValue resultRef="369">
              <real xmlns="http://schemas.mathsoft.com/math50">1</real>
            </secondTickValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="2">
          <axisLine position="origin" positionticmark="0" legendWidth="111.706666666667" />
          <axisGrid>
            <gridFrequency>3</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="378">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">K1</id>
                  <id xml:space="preserve">y</id>
                </apply>
              </math>
              <math resultRef="379">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="380">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">Kn</id>
                  <sequence>
                    <real>1</real>
                    <id xml:space="preserve">y</id>
                  </sequence>
                </apply>
              </math>
              <math resultRef="381">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="373">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <secondTickValue resultRef="375">
              <real xmlns="http://schemas.mathsoft.com/math50">1</real>
            </secondTickValue>
            <endValue resultRef="377">
              <real xmlns="http://schemas.mathsoft.com/math50">2</real>
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

7. Verwenden Sie die symbolische Auswertung, um die Beziehung zwischen den einzelnen Funktionen und ihrer skalierten Version zu zeigen:

8. Erstellen Sie ein Diagramm, um Folgendes zu zeigen:

<region id="ID0EVABAB" height="297.6" width="537.8685724213899" top="2995.2000000000003" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white" origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace resultRef="400">
          <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">Linien</traceStyle>
        </trace>
        <trace resultRef="401">
          <traceStyle color="#FF662D91" symbol="circle" line-weight="1" line-style="None">lines</traceStyle>
        </trace>
      </traces>
      <graph-size width="380.26857242139" height="240" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="5">
          <axisLine position="origin" positionticmark="0" legendWidth="60.74" />
          <axisGrid>
            <gridFrequency>6</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="404">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">y</id>
              </math>
              <math resultRef="405">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <secondTickValue resultRef="403">
              <real xmlns="http://schemas.mathsoft.com/math50">1</real>
            </secondTickValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="2">
          <axisLine position="origin" positionticmark="0" legendWidth="119.66" />
          <axisGrid>
            <gridFrequency>3</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <plotEquations>
            <plotEquation>
              <math resultRef="412">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <mult />
                  <apply>
                    <id labels="FUNCTION" xml:space="preserve">K0</id>
                    <id xml:space="preserve">y</id>
                  </apply>
                  <apply>
                    <pow />
                    <id xml:space="preserve">e</id>
                    <id xml:space="preserve">y</id>
                  </apply>
                </apply>
              </math>
              <math resultRef="413">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="414">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id labels="FUNCTION" xml:space="preserve">K0.sc</id>
                  <id xml:space="preserve">y</id>
                </apply>
              </math>
              <math resultRef="415">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="407">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <secondTickValue resultRef="409">
              <real xmlns="http://schemas.mathsoft.com/math50">1</real>
            </secondTickValue>
            <endValue resultRef="411">
              <real xmlns="http://schemas.mathsoft.com/math50">2</real>
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

Die modifizierten Bessel-Funktionen der zweiten Art haben keine Spitzenwerte.

Zugehörige Themen

Modifizierte (hyperbolische) Bessel-Funktionen

War dies hilfreich?