Example: Modified Bessel Functions of the First Kind

Functions > Special Functions > Bessel Functions > Example: Modified Bessel Functions of the First Kind

Show the relationship between functions I0, I1, and In. Also show the relationships between these functions and their scaled versions.

1. Define two step range variables:

2. Plot functions I0 and I1. Add the second order function In to the plot:

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                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">x</id>
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            <plotEquation>
              <math>
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                  <id labels="FUNCTION" xml:space="preserve">In</id>
                  <sequence>
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m</id>
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">x</id>
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3. Plot the fifth and eighth order function In:

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                      <real>3</real>
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• The higher the order of the In function, the sharper the transition from zero to infinity becomes.

• Only function I0 has its origin at (x=0,y=1). All other orders have their origins at (x=0,y=0).

4. Create a plot to show that I0(y)=In(0,y). Reset the tick mark values to zoom in the x-axis in order to show more details:

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5. Create a plot to show that I1(y)=In(1,y). Reset the tick mark values to zoom in the x-axis in order to show more details:

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6. Use symbolic evaluation to show the relationship between each function and its scaled version:

7. Use a plot to show that:

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The modified Bessel functions of the first kind have no peaks.