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Math Help - Bessel function

  1. #1
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    Bessel function

    This was very different and i need help please.

    Show that Jo (the Bessel function of order 0)

    satisfies the differential equation: x^2*Jo"(x) + x*Jo'(x) + x^2*Jo(x) = 0

    b.) evaluate 0_1 INT Jo(x) dx correct to three decimal places.

    thankyou for looking at this one.
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  2. #2
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    Quote Originally Posted by rcmango View Post
    This was very different and i need help please.

    Show that Jo (the Bessel function of order 0)

    satisfies the differential equation: x^2*Jo"(x) + x*Jo'(x) + x^2*Jo(x) = 0

    b.) evaluate 0_1 INT Jo(x) dx correct to three decimal places.

    thankyou for looking at this one.
    You had better tell us what definition of J_0 you want us to start from, one
    of the most common definitions is as a solution of the DE.

    RonL
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  3. #3
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    Quote Originally Posted by CaptainBlank View Post
    You had better tell us what definition of J_0 you want us to start from, one
    of the most common definitions is as a solution of the DE.

    RonL
    This happens to be a standard definition:
    "Bessel function of order zero of the first kind"

    J_0(x)= SUM(n=0,+oo) [ (-1)^n * x^{2n} ]/ [(n!)^2 * 2^{2n} ]
    (If I remember correctly.)
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  4. #4
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    Quote Originally Posted by ThePerfectHacker View Post
    This happens to be a standard definition:
    "Bessel function of order zero of the first kind"

    J_0(x)= SUM(n=0,+oo) [ (-1)^n * x^{2n} ]/ [(n!)^2 * 2^{2n} ]
    (If I remember correctly.)
    There is no standard definition you can define it as the solution of the DE
    with f(0)=1, f'(0)=0, you can define it as the power series, or as one of the integral forms:

    J_0(z) = (1/pi) integral_{0,pi} cos(z cos(theta)) dtheta

    or:

    J_0(z) = (1/pi) integral_{0,pi} cos(z sin(theta)) dtheta

    (check these I'm typing from memory here )

    Abramowitz and Stegun start with the DE then give the integral forms
    and some time later give the series.

    RonL

    RonL
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  5. #5
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    its alright, i don't think this this problem is the norm. I picked it because it was much different, and it looked hard. appreciate your help so far.
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  6. #6
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    Quote Originally Posted by rcmango View Post
    This was very different and i need help please.

    Show that Jo (the Bessel function of order 0)

    satisfies the differential equation: x^2*Jo"(x) + x*Jo'(x) + x^2*Jo(x) = 0
    You can do this using ImPerfectHacker's series representation for J_0
    differentiating term by term twice, and then plugging the resulting series
    into the DE.

    This can be done, and I have done it in a notebook here, but it is very
    fiddly and I definitly would not like to type it out here.

    It might be easier to work this in the oposite direction - find the series
    solution to the DE and show that it is ImPerfectHacker's series.

    RonL
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  7. #7
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    Quote Originally Posted by CaptainBlank View Post

    It might be easier to work this in the oposite direction - find the series
    solution to the DE and show that it is ImPerfectHacker's series.
    It is easier but I do not think the poster is familar with the "Frobenius Method". But anyways, that is not really the problem we can just assume an analytic solution exists at x=0 and just see what it becomes. Which is the Bessel series.

    J_0(z) = (1/pi) integral_{0,pi} cos(z cos(theta)) dtheta

    or:

    J_0(z) = (1/pi) integral_{0,pi} cos(z sin(theta)) dtheta
    That is a strange way to define them. There is not motivation behind it.
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  8. #8
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    Quote Originally Posted by ThePerfectHacker View Post
    It is easier but I do not think the poster is familar with the "Frobenius Method". But anyways, that is not really the problem we can just assume an analytic solution exists at x=0 and just see what it becomes. Which is the Bessel series.



    That is a strange way to define them. There is not motivation behind it.
    No idea I would have to dig out a copy Watson which I don't have
    access to anymore to find that.

    To me it looks a bit of a challenge just to prove that they define the
    same function.

    RonL
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  9. #9
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    "Frobenius Method"
    no, sorry not familiar with it.

    Ya appreciate the effort for this one. Its a more difficult problem than i anticipated.

    Thanks.
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  10. #10
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    Quote Originally Posted by rcmango View Post
    . Its a more difficult problem than i anticipated.

    Thanks.
    It is not difficult, it is just long. And because LaTeX is not working we do not want to post image uploads.
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