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Normalization of Bessel functions of the first kind

  1. Nov 20, 2013 #1

    ShayanJ

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    Gold Member

    Before stating the main question,which section should the special functions' questions be asked?

    Now consider the Bessel differential equation:

    [itex]
    \rho \frac{d^2}{d\rho^2}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})+\frac{d}{d\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})+(\frac{\alpha_{\nu m}^2 \rho}{a^2}-\frac{\nu^2}{\rho})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})=0
    [/itex]

    and a similar equation but with [itex] \alpha_{\nu m} [/itex] replaced by [itex] \alpha_{\nu n} [/itex] where [itex] \alpha_{\nu s} [/itex] is the [itex]s[/itex]th root of [itex] J_{\nu}(x)[/itex].

    Now if one multiplies the first equation by [itex] J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})[/itex] and the second by [itex]J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})[/itex] and then subtracts the second from the first,the following will be found upon integration of the whole equation from 0 to a:

    [itex]
    \int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})\frac{d}{d\rho}[\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})]d\rho-\int_0^a J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\frac{d}{d\rho}[\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})]d\rho=\frac{\alpha_{\nu n}^2-\alpha_{\nu m}^2}{a^2}\int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a}) \rho d\rho
    [/itex]

    Integrating the LHS by part and cancelling gives:

    [itex]
    J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})|_0^a-J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})|_0^a=\frac{\alpha_{\nu n}^2-\alpha_{\nu m}^2}{a^2}\int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a}) \rho d\rho
    [/itex]

    Using [itex] \frac{d}{dx}J_n(x)=\frac{n}{x}J_n(x)-J_{n+1}(x) [/itex]:

    [itex]
    J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})\rho[\frac{\nu a}{\alpha_{\nu m}\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})-J_{\nu+1}(\alpha_{\nu m}\frac{\rho}{a})]|_0^a-J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\rho[\frac{\nu a}{\alpha_{\nu n}\rho}J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})-J_{\nu+1}(\alpha_{\nu n}\frac{\rho}{a})]|_0^a=\frac{\alpha_{\nu n}^2-\alpha_{\nu m}^2}{a^2}\int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a}) \rho d\rho
    [/itex]

    After placing [itex]\alpha_{\nu n}=\alpha_{\nu m}+\varepsilon [/itex] and taking the limit as [itex] \varepsilon\rightarrow 0[/itex] and using [itex] \frac{d}{dx}J_n(x)=\frac{n}{x}J_n(x)-J_{n+1}(x) [/itex] to replace terms involving Js with [itex] \varepsilon [/itex] in their arguments and calculating the terms in the boundaries:
    [itex]
    -J_{\nu+1}(\alpha_{\nu m})\varepsilon[-aJ_{\nu+1}(\alpha_{\nu m})](\alpha_{\nu m})=\frac{2\alpha_{\nu m} \varepsilon}{a^2}\int_0^a J^2_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\rho d\rho
    [/itex]

    Which gives:

    [itex]
    \int_0^a [J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})]^2\rho d\rho=\frac{a^3}{2\alpha_{\nu m}}[J_{\nu+1}(\alpha_{\nu m})]^2
    [/itex]

    But the correct equation is:

    [itex]
    \int_0^a [J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})]^2\rho d\rho=\frac{a^2}{2}[J_{\nu+1}(\alpha_{\nu m})]^2
    [/itex]
    (This is what you find about normalization of Bessel functions everywhere)
    What's wrong in my calculations?
    Thanks
     
  2. jcsd
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