Functions > Special Functions > Bessel Functions > Example: Hankel Functions of the First and Second Kind
  
Example: Hankel Functions of the First and Second Kind
Show the relationship between functions H1 and H2 with functions Jn and Yn. Also show the relationships between these functions and their scaled versions.
The Hankel functions are not defined at 0.
1. Use symbolic evaluation to show the relationship between H1, Jn, and Yn:
Click to copy this expression
2. Use symbolic evaluation to show the relationship between H2, Jn, and Yn:
Click to copy this expression
◦ The real components of H1 and H2 are identical in value and sign.
◦ The imaginary components of H1 and H2 are identical in value but opposite in sign.
3. Note these two other important relationships between H1 and H2:
Click to copy this expression
Click to copy this expression
4. Create a plot to visually show that |H1|=|H2|:
Click to copy this expression
Click to copy this expression
Click to copy this expression
5. Create a plot to show the relationship between H1, Jn and Yn:
Click to copy this expression
6. Use symbolic evaluation to show the relationship between the Hankel functions and their scaled version:
Click to copy this expression
Click to copy this expression
7. Create a plot to show that:
Click to copy this expression
Click to copy this expression
8. Create a plot to show that:
Click to copy this expression
Click to copy this expression