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Delta derivative identity

  1. Dec 16, 2009 #1
    Hello.

    1. The problem statement, all variables and given/known data

    I would like to solve the following:

    [tex]
    \[\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,f\left( x \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left[ {a\left( {x - x_0 } \right)} \right]} \]
    [/tex]

    The solution I found in a paper is:

    [tex]
    \[\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,f\left( x \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left[ {a\left( {x - x_0 } \right)} \right]} = - a^{ - 2} \frac{{\rm{d}}}{{{\rm{d}}x}}f\left( {x_0 } \right)\]
    [/tex]

    Also, there's a similar expression http://functions.wolfram.com/GeneralizedFunctions/DiracDelta/20/ShowAll.html" [Broken] (check the last equation).

    2. Relevant equations

    [tex]
    \[
    \delta \left[ {a\left( {x - x_0 } \right)} \right] = \left| a \right|^{ - 1} \delta \left( {x - x_0 } \right)
    \]
    [/tex]

    [tex]
    \[\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,f\left( x \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left( {x - x_0 } \right)} = - \frac{{\rm{d}}}{{{\rm{d}}x}}f\left( {x_0 } \right)\]
    [/tex]

    3. The attempt at a solution

    Representing the delta as a Dirac sequence and integrating by parts:

    [tex]
    \[
    \begin{array}{l}
    \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left[ {a\left( {x - x_0 } \right)} \right]f\left( x \right)} = \mathop {\lim }\limits_{k \to \infty } \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\frac{{\rm{d}}}{{{\rm{d}}x}}\psi _k^{} \left[ {a\left( {x - x_0 } \right)} \right]f\left( x \right)} \\
    & = \left\{ \begin{array}{l}
    u = f\left( x \right) & {\rm{d}}u = \frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right){\rm{d}}x \\
    {\rm{d}}v = \frac{{\rm{d}}}{{{\rm{d}}x}}\psi _k^{} \left[ {a\left( {x - x_0 } \right)} \right]{\rm{d}}x & v = \psi _k \left[ {a\left( {x - x_0 } \right)} \right] \\
    \end{array} \right. \\
    & = \mathop {\lim }\limits_{k \to \infty } \left\{ {\left. {f\left( x \right)\psi _k \left[ {a\left( {x - x_0 } \right)} \right]} \right|_{ - \infty }^\infty - \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\psi _k \left[ {a\left( {x - x_0 } \right)} \right]\frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right)} } \right\} \\
    & = - \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\delta \left[ {a\left( {x - x_0 } \right)} \right]\frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right)} \\
    & = - \left| a \right|^{ - 1} \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\delta \left( {x - x_0 } \right)\frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right)} \\
    & = - \left| a \right|^{ - 1} \frac{{{\rm{d}}f}}{{{\rm{d}}x}}\left( {x_0 } \right) \\
    \end{array}
    \]
    [/tex]

    Thank you for your time!
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
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