Пример. Функции Кельвина

Функции > Специальные функции > Функции Бесселя > Пример. Функции Кельвина

Покажите взаимосвязь между функциями ber и bei. Эти функции не имеют масштабированных форм.

1. Постройте график функций ber и bei нулевого порядка:

Щелкните для копирования этого выражения

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Две функции кажутся симметричными.

2. Сбросьте значение делений, чтобы увеличить масштаб оси X и показать, как два пика функций ber и bei изменяются с увеличением порядка:

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3. Используйте аналитические преобразования, чтобы показать взаимосвязи между функциями ber и bei с функцией Jn, а затем создайте графики, чтобы показать результаты визуально.

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Этот пример демонстрирует функцию ber, но те же выводы применимы к функции bei.

4. Определите вектор Z1 с помощью функции ber:

Для функций localmin и localmax требуется входная матрица из двух столбцов. Передайте вектор Z в функцию augment, чтобы создать такую матрицу.

5. Найдите локальные минимумы функции ber:

Функция ber имеет 8 минимумов, когда m=0. Функция симметрична. У функций более высокого порядка имеется различное число минимумов с другими положениями и амплитудами.

6. Найдите локальные максимумы функции ber:

Функция ber имеет 7 пиков, когда m=0. Функция симметрична. У функций более высокого порядка имеется различное число максимумов с другими положениями и амплитудами.

7. Найдите вертикальные координаты четырех минимумов в правой части.

8. Используйте функцию match, чтобы найти индекс каждого соответствующего минимума, а затем найдите эквивалентные горизонтальные координаты.

9. Найдите вертикальные координаты четырех максимумов в правой части.

10. Используйте функцию match, чтобы найти индекс каждого соответствующего максимума, а затем найдите эквивалентные горизонтальные координаты.

Диапазон амплитуд максимумов и минимумов значительно изменяется, и в результате трудно представить все на одном графике.

11. Используйте маркеры, чтобы показать первые минимумы (D0) и первые максимумы (P0):

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          <axisGrid>
            <gridFrequency>3</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers>
            <marker resultRef="670" display="true" color="#FF2E3192" line-style="Dash" value="6">
              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">xd0</id>
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              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">xp0</id>
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          <plotEquations>
            <plotEquation>
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                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">s0</id>
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                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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              <real xmlns="http://schemas.mathsoft.com/math50">10</real>
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        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-10" end="2">
          <axisLine position="origin" positionticmark="1" legendWidth="119.073333333333" />
          <axisGrid>
            <gridFrequency>7</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers>
            <marker resultRef="682" display="true" color="#FF2E3192" line-style="Dash" value="-8.8583159660450352">
              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">D0</id>
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            <marker resultRef="684" display="true" color="#FF068149" line-style="Dot" value="1">
              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">P0</id>
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          </markers>
          <plotEquations>
            <plotEquation>
              <math resultRef="692">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <id xml:space="preserve">ber</id>
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                    <real>0</real>
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              </math>
              <math resultRef="693">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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              <real xmlns="http://schemas.mathsoft.com/math50">2</real>
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Как ожидалось, график демонстрирует вторые минимумы слева от оси Y.

12. Используйте маркеры, чтобы показать следующие минимумы (D1) и следующие максимумы (P1):

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    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
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        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-20" end="20">
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          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers>
            <marker resultRef="719" display="true" color="#FF2E3192" line-style="Dash" value="15">
              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">xd1</id>
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            <marker resultRef="721" display="true" color="#FF068149" line-style="Dash" value="10.5">
              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">xp1</id>
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          </markers>
          <plotEquations>
            <plotEquation>
              <math resultRef="729">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">s1</id>
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              <math resultRef="730">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
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              <real xmlns="http://schemas.mathsoft.com/math50">-20</real>
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        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-3000" end="200">
          <axisLine position="origin" positionticmark="2" legendWidth="119.073333333333" />
          <axisGrid>
            <gridFrequency>17</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers>
            <marker resultRef="731" display="true" color="#FF068149" line-style="Dash" value="153.7685819660004">
              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">P1</id>
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            <marker resultRef="733" display="true" color="#FF2E3192" line-style="Dash" value="-2967.2545346351285">
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          <plotEquations>
            <plotEquation>
              <math resultRef="741">
                <apply xmlns="http://schemas.mathsoft.com/math50">
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                  <sequence>
                    <real>0</real>
                    <id xml:space="preserve">s1</id>
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                </apply>
              </math>
              <math resultRef="742">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
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          <xyDomain scale-type="linear" auto-scale="true">
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              <real xmlns="http://schemas.mathsoft.com/math50">-3000</real>
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            <endValue resultRef="740">
              <real xmlns="http://schemas.mathsoft.com/math50">200</real>
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</region>

Как ожидалось, график демонстрирует вторые минимумы и вторые максимумы слева от оси Y.

13. Используйте маркеры, чтобы показать следующие минимумы (D2) и следующие максимумы (P2):

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  <plot background-type="white" origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
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      <traces>
        <trace resultRef="767">
          <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
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      </traces>
      <graph-size width="418.666666666666" height="230.4" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-25" end="25">
          <axisLine position="origin" positionticmark="4" legendWidth="66.7833333333333" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers>
            <marker resultRef="768" display="true" color="#FF2E3192" line-style="Dash" value="23.9">
              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">xd2</id>
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            <marker resultRef="770" display="true" color="#FF068149" line-style="Dash" value="19.4">
              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">xp2</id>
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          <plotEquations>
            <plotEquation>
              <math resultRef="778">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">s2</id>
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              <math resultRef="779">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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          <axisGrid>
            <gridFrequency>6</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers>
            <marker resultRef="780" display="true" color="#FF068149" line-style="Dash" value="60152.318446411562">
              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">P2</id>
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            <marker resultRef="782" display="true" color="#FF2E3192" line-style="Dash" value="-1252953.921668866">
              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">D2</id>
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            <plotEquation>
              <math resultRef="790">
                <apply xmlns="http://schemas.mathsoft.com/math50">
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              <math resultRef="791">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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          <xyDomain scale-type="linear" auto-scale="true">
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              <real xmlns="http://schemas.mathsoft.com/math50">-1300000</real>
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              <real xmlns="http://schemas.mathsoft.com/math50">-1000000</real>
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Как ожидалось, график демонстрирует вторые минимумы и вторые максимумы слева от оси Y.

14. Используйте маркеры, чтобы показать следующие минимумы (D3) и следующие максимумы (P3):

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            <gridLabels display="true" />
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            <tickMarks display="true" />
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          <axisLabel />
          <markers>
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              <math resultRef="825">
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            <tickMarks display="true" />
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          <axisLabel />
          <markers>
            <marker resultRef="827" display="true" color="#FF068149" line-style="Dash" value="26606074.186449833">
              <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">P3</id>
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              <math resultRef="835">
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Как ожидалось, график демонстрирует вторые максимумы слева от оси Y.

Минимумы (D3) и их эквиваленты на левой стороне попадают точно на края диапазона, поэтому их лучше игнорировать.

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