Friday, 21 March 2014

special functions - Integration error spotting

What have I done wrong? 




I have to evaluate the following integral:



$$\int\limits_0^\infty\int\limits_0^{2\pi} \phi(r^2)\delta'(r^2-a)\delta\left(\theta- \left(n+{1\over 2}\right){\pi\over 2}\right) r \; d\theta \;dr$$



My (wrong) working:



$\implies 4\int\limits_0^\infty \phi(x)\delta'(x-a){1\over 2}dx$ by letting $x=r^2$



$\implies -2\phi'(a)$




I should be getting the integral equalling $\phi(a)-\phi'(a)$.



Thank you.

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