ZStardust
Dec16-09, 06:19 AM
Hello.
1. The problem statement, all variables and given/known data
I would like to solve the following:
\[\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,f\left( x \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left[ {a\left( {x - x_0 } \right)} \right]} \]
The solution I found in a paper is:
\[\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,f\left( x \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left[ {a\left( {x - x_0 } \right)} \right]} = - a^{ - 2} \frac{{\rm{d}}}{{{\rm{d}}x}}f\left( {x_0 } \right)\]
Also, there's a similar expression here (http://functions.wolfram.com/GeneralizedFunctions/DiracDelta/20/ShowAll.html) (check the last equation).
2. Relevant equations
\[
\delta \left[ {a\left( {x - x_0 } \right)} \right] = \left| a \right|^{ - 1} \delta \left( {x - x_0 } \right)
\]
\[\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,f\left( x \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left( {x - x_0 } \right)} = - \frac{{\rm{d}}}{{{\rm{d}}x}}f\left( {x_0 } \right)\]
3. The attempt at a solution
Representing the delta as a Dirac sequence and integrating by parts:
\[
\begin{array}{l}
\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left[ {a\left( {x - x_0 } \right)} \right]f\left( x \right)} = \mathop {\lim }\limits_{k \to \infty } \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\frac{{\rm{d}}}{{{\rm{d}}x}}\psi _k^{} \left[ {a\left( {x - x_0 } \right)} \right]f\left( x \right)} \\
& = \left\{ \begin{array}{l}
u = f\left( x \right) & {\rm{d}}u = \frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right){\rm{d}}x \\
{\rm{d}}v = \frac{{\rm{d}}}{{{\rm{d}}x}}\psi _k^{} \left[ {a\left( {x - x_0 } \right)} \right]{\rm{d}}x & v = \psi _k \left[ {a\left( {x - x_0 } \right)} \right] \\
\end{array} \right. \\
& = \mathop {\lim }\limits_{k \to \infty } \left\{ {\left. {f\left( x \right)\psi _k \left[ {a\left( {x - x_0 } \right)} \right]} \right|_{ - \infty }^\infty - \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\psi _k \left[ {a\left( {x - x_0 } \right)} \right]\frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right)} } \right\} \\
& = - \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\delta \left[ {a\left( {x - x_0 } \right)} \right]\frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right)} \\
& = - \left| a \right|^{ - 1} \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\delta \left( {x - x_0 } \right)\frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right)} \\
& = - \left| a \right|^{ - 1} \frac{{{\rm{d}}f}}{{{\rm{d}}x}}\left( {x_0 } \right) \\
\end{array}
\]
Thank you for your time!
1. The problem statement, all variables and given/known data
I would like to solve the following:
\[\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,f\left( x \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left[ {a\left( {x - x_0 } \right)} \right]} \]
The solution I found in a paper is:
\[\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,f\left( x \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left[ {a\left( {x - x_0 } \right)} \right]} = - a^{ - 2} \frac{{\rm{d}}}{{{\rm{d}}x}}f\left( {x_0 } \right)\]
Also, there's a similar expression here (http://functions.wolfram.com/GeneralizedFunctions/DiracDelta/20/ShowAll.html) (check the last equation).
2. Relevant equations
\[
\delta \left[ {a\left( {x - x_0 } \right)} \right] = \left| a \right|^{ - 1} \delta \left( {x - x_0 } \right)
\]
\[\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,f\left( x \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left( {x - x_0 } \right)} = - \frac{{\rm{d}}}{{{\rm{d}}x}}f\left( {x_0 } \right)\]
3. The attempt at a solution
Representing the delta as a Dirac sequence and integrating by parts:
\[
\begin{array}{l}
\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left[ {a\left( {x - x_0 } \right)} \right]f\left( x \right)} = \mathop {\lim }\limits_{k \to \infty } \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\frac{{\rm{d}}}{{{\rm{d}}x}}\psi _k^{} \left[ {a\left( {x - x_0 } \right)} \right]f\left( x \right)} \\
& = \left\{ \begin{array}{l}
u = f\left( x \right) & {\rm{d}}u = \frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right){\rm{d}}x \\
{\rm{d}}v = \frac{{\rm{d}}}{{{\rm{d}}x}}\psi _k^{} \left[ {a\left( {x - x_0 } \right)} \right]{\rm{d}}x & v = \psi _k \left[ {a\left( {x - x_0 } \right)} \right] \\
\end{array} \right. \\
& = \mathop {\lim }\limits_{k \to \infty } \left\{ {\left. {f\left( x \right)\psi _k \left[ {a\left( {x - x_0 } \right)} \right]} \right|_{ - \infty }^\infty - \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\psi _k \left[ {a\left( {x - x_0 } \right)} \right]\frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right)} } \right\} \\
& = - \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\delta \left[ {a\left( {x - x_0 } \right)} \right]\frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right)} \\
& = - \left| a \right|^{ - 1} \int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,\delta \left( {x - x_0 } \right)\frac{{\rm{d}}}{{{\rm{d}}x}}f\left( x \right)} \\
& = - \left| a \right|^{ - 1} \frac{{{\rm{d}}f}}{{{\rm{d}}x}}\left( {x_0 } \right) \\
\end{array}
\]
Thank you for your time!