Example: Polynomial Functions

Functions > Special Functions > Other Special Functions > Example: Polynomial Functions

Hermite Polynomial

1. Use the Her function to plot the values of the Hermite polynomial at different values of x.

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 <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-30" end="30">
 <axisLine position="origin" positionticmark="1" legendWidth="124.143333333333" />
 <axisGrid>
 <gridFrequency>7</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <plotEquations>
 <plotEquation>
 <math resultRef="260">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Her</id>
 <sequence>
 <real>1</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript>
 </id>
 </sequence>
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 <math resultRef="261">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="262">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Her</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="263">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="264">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Her</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="265">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="266">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Her</id>
 <sequence>
 <real>4</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="267">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 </plotEquations>
 <xyDomain scale-type="linear" auto-scale="true">
 <startValue resultRef="255">
 <real xmlns="http://schemas.mathsoft.com/math50">-30</real>
 </startValue>
 <secondTickValue resultRef="257">
 <real xmlns="http://schemas.mathsoft.com/math50">-20</real>
 </secondTickValue>
 <endValue resultRef="259">
 <real xmlns="http://schemas.mathsoft.com/math50">30</real>
 </endValue>
 </xyDomain>
 </yAxis>
 </axes>
 </xyPlot>
 </plot>
</region>

Laguerre Polynomial

1. Use the Lag function to plot the values of the Laguerre polynomial at different values of x.

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 <plotEquation>
 <math resultRef="295">
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 <real xmlns="http://schemas.mathsoft.com/math50">5</real>
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 <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-4" end="4">
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 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
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 <axisLabel />
 <markers />
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 <plotEquation>
 <math resultRef="303">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Lag</id>
 <sequence>
 <real>1</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="304">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="305">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Lag</id>
 <sequence>
 <real>2</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="306">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="307">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Lag</id>
 <sequence>
 <real>3</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="308">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="309">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Lag</id>
 <sequence>
 <real>4</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="310">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
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 <startValue resultRef="298">
 <real xmlns="http://schemas.mathsoft.com/math50">-4</real>
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 <secondTickValue resultRef="300">
 <real xmlns="http://schemas.mathsoft.com/math50">-2</real>
 </secondTickValue>
 <endValue resultRef="302">
 <real xmlns="http://schemas.mathsoft.com/math50">4</real>
 </endValue>
 </xyDomain>
 </yAxis>
 </axes>
 </xyPlot>
 </plot>
</region>

Legendre Polynomial

1. Use the Leg function to plot the values of the Legendre polynomial at different values of x.

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 <gridLabels display="true" />
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 <plotEquation>
 <math resultRef="339">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
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 <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-1" end="1">
 <axisLine position="origin" positionticmark="1" legendWidth="122.136666666667" />
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 <gridFrequency>3</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <plotEquations>
 <plotEquation>
 <math resultRef="347">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Leg</id>
 <sequence>
 <real>1</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="348">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="349">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Leg</id>
 <sequence>
 <real>2</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="350">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="351">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Leg</id>
 <sequence>
 <real>3</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="352">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="353">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Leg</id>
 <sequence>
 <real>4</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="354">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
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 </secondTickValue>
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 <real xmlns="http://schemas.mathsoft.com/math50">1</real>
 </endValue>
 </xyDomain>
 </yAxis>
 </axes>
 </xyPlot>
 </plot>
</region>

Chebyshev Polynomial of the First and Second Kind

1. Use the Tcheb function to plot the values of the Chebyshev polynomial of the first kind at different values of x.

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 <id xml:space="preserve">Tcheb</id>
 <sequence>
 <real>1</real>
 <id xml:space="preserve">
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 <math resultRef="381">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Tcheb</id>
 <sequence>
 <real>2</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="382">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="383">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Tcheb</id>
 <sequence>
 <real>3</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="384">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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 <plotEquation>
 <math resultRef="385">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Tcheb</id>
 <sequence>
 <real>4</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
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 <math resultRef="386">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
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 <endValue>
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 </endValue>
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 </yAxis>
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 </xyPlot>
 </plot>
</region>

2. Use the Ucheb function to plot the values of the Chebyshev polynomial of the second kind at different values of x.

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 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
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 <math resultRef="414">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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 <plotEquation>
 <math resultRef="415">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Ucheb</id>
 <sequence>
 <real>2</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
 </sequence>
 </apply>
 </math>
 <math resultRef="416">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="417">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id xml:space="preserve">Ucheb</id>
 <sequence>
 <real>3</real>
 <id xml:space="preserve">
 x<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
 </sequence>
 </apply>
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Jacobi Polynomial

1. Use the Jac function to plot the values of the Jacobi polynomial, with parameters a and b, at different values of x.

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 <sequence>
 <real>2</real>
 <id xml:space="preserve">a</id>
 <id xml:space="preserve">b</id>
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 <sequence>
 <real>3</real>
 <id xml:space="preserve">a</id>
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 <sequence>
 <real>4</real>
 <id xml:space="preserve">a</id>
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 <id xml:space="preserve">
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