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Is the following conclusion correct? $$\delta(x)\delta(y)=\delta(x^2+y^2)$$ where $\delta$ is dirac delta function. Please look equation 42-44 in https://mathworld.wolfram.com/DeltaFunction.html

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    The left hand side is the correct two-dimensional Delta function around (0,0) in Cartesian coordinates. Now this can indeed be written in terms of polar coordinates, however your right hand side must then be divided by $\pi$. See also equation (46) in MathWorld. – M. Wind Jun 13 '20 at 05:56
  • Hi Wind, Thank you, that makes sense. However, I am wondering if you can advise me on how to prove it? Thanks – Bita Jun 13 '20 at 12:05
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    The Dirac delta function is best understood through its properties on integration. Therefore start with the integral over the xy-plane of the left hand side, with element dxdy. Now make the transition to polar coordinates. The surface element becomes rdrd$\phi$. Integration over $\phi$ yields 2$\pi$. Note that rdr may also be written as 0.5d(r^2). You can now formulate an appropriate version of the delta function on the right hand side, so that the new integral yields the same value as the one in Cartesian coordinates. – M. Wind Jun 13 '20 at 17:20
  • Thank you, you helped a lot! – Bita Jun 13 '20 at 19:46

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