Weber differential equation

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Main Question or Discussion Point

Abramovitz presents even and odd solutions to the Weber equation.

He also presents standard solutions as a pair of parabolic cylinder functions.
Clearly any linear combination of the even and odd solutions is also a solution of the equation.

My question is: Why is the parabolic cylinder function so special to be considered a "standard" solution?
 
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  • #2
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Abramovitz presents even and odd solutions to the Weber equation.

He also presents standard solutions as a pair of parabolic cylinder functions.
Clearly any linear combination of the even and odd solutions is also a solution of the equation.

My question is: Why is the parabolic cylinder function so special to be considered a "standard" solution?
Can you write this solution?
 
  • #3
191
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Weber equation
[tex]
\frac{d^2y}{dx^2}-(x^2/4+a)y=0
[/tex]

Even solution
[tex]
y_1=e^{-x^2/2}M(\frac{a}{2}+\frac{1}{4},\frac{1}{2},\frac{x^2}{2})
[/tex]

Odd solution
[tex]
y_2=xe^{x^2/2}M(-\frac{a}{2}+\frac{1}{4},\frac{1}{2},-\frac{x^2}{2})
[/tex]

where M is the Kummer function.

Independent parabolic cylinder functions

[tex]D_\nu(x)[/tex] and [tex]D_{-\nu-1}(ix)[/tex]
 

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