Пример: Гамма-функции

Функции > Специальные функции > Другие специальные функции > Пример: Гамма-функции

1. Используйте функцию Γ, чтобы построить график гамма-функции Эйлера для различных значений x.

Щелкните для копирования этого выражения

<region id="ID0ELUIAB" top="115.20000000000002" left="38.400000000000006" height="288" width="576.59999999999991" 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="220">
 <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
 </trace>
 </traces>
 <graph-size width="457.4" height="230.4" />
 <axes>
 <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-5" end="5">
 <axisLine position="origin" positionticmark="3" legendWidth="66.5166666666667" />
 <axisGrid>
 <gridFrequency>11</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers>
 <marker resultRef="221" display="true" color="#FF000000" line-style="Dot" value="-1">
 <real xmlns="http://schemas.mathsoft.com/math50">-1</real>
 </marker>
 <marker resultRef="223" display="true" color="#FF000000" line-style="Dot" value="-2">
 <real xmlns="http://schemas.mathsoft.com/math50">-2</real>
 </marker>
 <marker resultRef="225" display="true" color="#FF000000" line-style="Dot" value="-3">
 <real xmlns="http://schemas.mathsoft.com/math50">-3</real>
 </marker>
 <marker resultRef="227" display="true" color="#FF000000" line-style="Dot" value="0">
 <real xmlns="http://schemas.mathsoft.com/math50">0</real>
 </marker>
 <marker resultRef="229" display="true" color="#FF000000" line-style="Dot" value="-4">
 <real xmlns="http://schemas.mathsoft.com/math50">-4</real>
 </marker>
 <marker display="true" color="#FF000000" line-style="Dot" value="-5" />
 </markers>
 <plotEquations>
 <plotEquation>
 <math resultRef="235">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript>
 </id>
 </math>
 <math resultRef="236">
 <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="232">
 <real xmlns="http://schemas.mathsoft.com/math50">-4</real>
 </secondTickValue>
 <endValue resultRef="234">
 <real xmlns="http://schemas.mathsoft.com/math50">5</real>
 </endValue>
 </xyDomain>
 </xAxis>
 <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-15" end="15">
 <axisLine position="origin" positionticmark="5" legendWidth="90.34" />
 <axisGrid>
 <gridFrequency>7</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <plotEquations>
 <plotEquation>
 <math resultRef="243">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id labels="FUNCTION" xml:space="preserve">Γ</id>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript>
 </id>
 </apply>
 </math>
 <math resultRef="244">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 </plotEquations>
 <xyDomain scale-type="linear" auto-scale="true">
 <startValue resultRef="238">
 <real xmlns="http://schemas.mathsoft.com/math50">-15</real>
 </startValue>
 <secondTickValue resultRef="240">
 <real xmlns="http://schemas.mathsoft.com/math50">-10</real>
 </secondTickValue>
 <endValue resultRef="242">
 <real xmlns="http://schemas.mathsoft.com/math50">15</real>
 </endValue>
 </xyDomain>
 </yAxis>
 </axes>
 </xyPlot>
 </plot>
</region>

Функция является аналитической везде, кроме точек 0, -1, -2... и т. д.

2. Используйте неполную функцию Γ, чтобы построить график неполной гамма-функции для различных значений x.

<region id="ID0EVWIAB" top="537.60000000000014" left="38.400000000000006" height="307.2" width="595.80000000000007" 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="261">
 <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
 </trace>
 <trace resultRef="262">
 <traceStyle color="#FF068149" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
 </trace>
 <trace resultRef="263">
 <traceStyle color="#FF2E3192" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
 </trace>
 </traces>
 <graph-size width="457.4" height="230.4" />
 <axes>
 <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-1" end="10">
 <axisLine position="origin" positionticmark="0" legendWidth="66.5166666666667" />
 <axisGrid>
 <gridFrequency>12</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <plotEquations>
 <plotEquation>
 <math resultRef="270">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript>
 </id>
 </math>
 <math resultRef="271">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 </plotEquations>
 <xyDomain scale-type="linear" auto-scale="true">
 <startValue resultRef="265">
 <real xmlns="http://schemas.mathsoft.com/math50">-1</real>
 </startValue>
 <secondTickValue resultRef="267">
 <real xmlns="http://schemas.mathsoft.com/math50">0</real>
 </secondTickValue>
 <endValue resultRef="269">
 <real xmlns="http://schemas.mathsoft.com/math50">10</real>
 </endValue>
 </xyDomain>
 </xAxis>
 <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="2.5">
 <axisLine position="origin" positionticmark="1" legendWidth="106.69" />
 <axisGrid>
 <gridFrequency>6</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <plotEquations>
 <plotEquation>
 <math resultRef="278">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id labels="FUNCTION" xml:space="preserve">Γ</id>
 <sequence>
 <real>1</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="279">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="280">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id labels="FUNCTION" xml:space="preserve">Γ</id>
 <sequence>
 <real>2</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="281">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="282">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id labels="FUNCTION" xml:space="preserve">Γ</id>
 <sequence>
 <real>3</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="283">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 </plotEquations>
 <xyDomain scale-type="linear" auto-scale="true">
 <startValue resultRef="273">
 <real xmlns="http://schemas.mathsoft.com/math50">0</real>
 </startValue>
 <secondTickValue resultRef="275">
 <real xmlns="http://schemas.mathsoft.com/math50">0.5</real>
 </secondTickValue>
 <endValue resultRef="277">
 <real xmlns="http://schemas.mathsoft.com/math50">2.5</real>
 </endValue>
 </xyDomain>
 </yAxis>
 </axes>
 </xyPlot>
 </plot>
</region>

3. Используйте функцию lnΓ, чтобы построить график натурального логарифма гамма-функции Эйлера для различных значений x.

<region id="ID0E2YIAB" top="979.20000000000016" left="38.400000000000006" height="288" width="577" 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="300">
 <traceStyle color="#FFED1D2F" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
 </trace>
 </traces>
 <graph-size width="438.6" height="230.4" />
 <axes>
 <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-1" end="5">
 <axisLine position="origin" positionticmark="1" legendWidth="66.5166666666667" />
 <axisGrid>
 <gridFrequency>7</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <plotEquations>
 <plotEquation>
 <math resultRef="307">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript>
 </id>
 </math>
 <math resultRef="308">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 </plotEquations>
 <xyDomain scale-type="linear" auto-scale="true">
 <startValue resultRef="302">
 <real xmlns="http://schemas.mathsoft.com/math50">-1</real>
 </startValue>
 <secondTickValue resultRef="304">
 <real xmlns="http://schemas.mathsoft.com/math50">0</real>
 </secondTickValue>
 <endValue resultRef="306">
 <real xmlns="http://schemas.mathsoft.com/math50">5</real>
 </endValue>
 </xyDomain>
 </xAxis>
 <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-1" end="6">
 <axisLine position="origin" positionticmark="1" legendWidth="103.32" />
 <axisGrid>
 <gridFrequency>8</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <plotEquations>
 <plotEquation>
 <math resultRef="315">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id labels="FUNCTION" xml:space="preserve">lnΓ</id>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript>
 </id>
 </apply>
 </math>
 <math resultRef="316">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 </plotEquations>
 <xyDomain scale-type="linear" auto-scale="true">
 <startValue resultRef="310">
 <real xmlns="http://schemas.mathsoft.com/math50">-1</real>
 </startValue>
 <secondTickValue resultRef="312">
 <real xmlns="http://schemas.mathsoft.com/math50">0</real>
 </secondTickValue>
 <endValue resultRef="314">
 <real xmlns="http://schemas.mathsoft.com/math50">6</real>
 </endValue>
 </xyDomain>
 </yAxis>
 </axes>
 </xyPlot>
 </plot>
</region>

Похожие темы

Гамма-функции

Было ли это полезно?