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Weber differential equation

  1. Apr 3, 2010 #1
    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?
     
    Last edited: Apr 3, 2010
  2. jcsd
  3. Apr 4, 2010 #2
    Can you write this solution?
     
  4. Apr 4, 2010 #3
    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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